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Research ArticleBlow-Up of Solutions for a Class of Nonlinear PseudoparabolicEquations with a Memory Term
Huafei Di1 and Yadong Shang12
1 School of Mathematics and Information Science Guangzhou University Guangzhou Guangdong 510006 China2 Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes Guangzhou UniversityGuangzhou Guangdong 510006 China
Correspondence should be addressed to Yadong Shang gzydshang126com
Received 7 June 2014 Accepted 1 July 2014 Published 4 August 2014
Academic Editor BoQing Dong
Copyright copy 2014 H Di and Y Shang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We consider the nonlinear pseudoparabolic equation with a memory term 119906119905minusΔ119906minusΔ119906
119905+int
119905
0120582 (119905 minus 120591) Δ119906 (120591) 119889120591 = div (|nabla119906|119901minus2119906)+
1199061+120572 119909 isin Ω 119905 gt 0 with an initial condition andDirichlet boundary condition Under negative initial energy and suitable conditions
on p 120572 and the relaxation function 120582(119905) we prove a finite-time blow-up result by using the concavity method
1 Introduction
In this paper we consider the initial boundary value problemfor a class of nonlinear pseudoparabolic equations with amemory term
119906119905minus 120573Δ119906 minus 120574Δ119906
119905+ int
119905
0
120582 (119905 minus 120591) Δ119906 (119909 120591) 119889120591
= 120575 div (|nabla119906|119901minus2nabla119906) + 119891 (119906) 119909 isin Ω 119905 gt 0
119906 (119909 0) = 1199060(119909) 119909 isin Ω
119906(119909 119905)|120597Ω
= 0 119909 isin 120597Ω 119905 gt 0
(1)
where Ω sub R119899 is a bounded domain with a smoothboundary 120597Ω 120582 R
+rarr R is a given continuous
function 120573 120574 and 120575 are all real constant parameters 119901 gt 2and div(|nabla119906|119901minus2nabla119906) is the so-called 119901-Laplace operator Thistype of equations describes a variety of important physicalprocesses such as the analysis of heat conduction inmaterialswithmemory electric signals in nonlinear telegraph line withnonlinear damping viscous flow in materials with memory[1] vibration of nonlinear elastic rod with viscosity [2]nonlinear bidirectional shallow water waves [3] and thevelocity evolution of ion-acoustic waves in a collisionlessplasma when an ion viscosity is invoked [4]
Equation (1) includes many important mathematicalphysics models
In the absence of thememory term the viscous term and119901-Laplace operator term (120574 = 120575 = 0 120582(119904) = 0) 120573 = 1 themodel reduces to semilinear parabolic equation
119906119905minus Δ119906 = 119891 (119906) 119909 isin Ω 119905 gt 0 (2)
On the existence nonexistence and the properties of solu-tions of (2) there have been many results [5ndash9]
In the absence of the memory term and 119901-Laplaceoperator term (120575 = 0 120582(119904) = 0) 120573 = 120574 = 1 the modelreduces to semilinear pseudoparabolic equation
119906119905minus Δ119906 minus Δ119906
119905= 119891 (119906) 119909 isin Ω 119905 gt 0 (3)
Kaikina et al [10] discussed the periodic boundary valueproblem of (3) under some assumption forms of nonlinearfunction 119891 Cao et al [11] investigated a class of periodicproblems of pseudoparabolic type equations with nonlinearperiodic sources A rather complete classification of theexponent 119901 was given in terms of the existence and nonexis-tence of nontrivial and nonnegative periodic solutions Caoet al [12] dealt with the Cauchy problem for semilinearpseudoparabolic equations Existenceand uniqueness of local
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 507494 7 pageshttpdxdoiorg1011552014507494
2 Abstract and Applied Analysis
solutions were proved and the large-time behavior wasinvestigated Kaikina [13] and Xu and Su [14] discussedthe initial boundary value problems of pseudoparabolicequation (3) under some classes of nonlinear function 119891(119906)They obtained some sufficient conditions of existence anduniqueness of local solutions and the large-time behavior ofglobal solutions
In the absence of the memory term and the viscous term(120582(119904) = 0 120574 = 0) 120573 = 0 120575 = 1 (1) becomes nonlinearparabolic equation with 119901-Laplace nonlinear term
119906119905= div (|nabla119906|119901minus2nabla119906) + 119891 (119906) 119909 isin Ω 119905 gt 0 (4)
Tsutsumi [15] studied the initial boundary value problem of(4) with 119891(119906) = 119906
1+120572 where 119901 lt 2 + 120572 He obtained theexistence of global weak solutions by using the potential wellmethod Liu and Zhao [16] considered the same problemwith critical initial conditions 119869(119906
0) = 119889 or 119868(119906
0) = 0 and
proved the existence of global solution for this problem Xuet al [17] discussed (4) at the high energy level where 119901 lt
2 + 120572 lt infin if 119899 le 119901 and 119901 lt 2 + 120572 le 119899119901(119899 minus 119901) if119899 gt 119901They proved the finite time blow-up of solutions by thecomparison principle and variational methods Messaoudi in[18] considers an initial boundary value problem related to (4)and proves under suitable conditions on 119891 a blow-up resultfor solutions with vanishing or negative initial energy
In the absence of the viscous term and119901-Laplace operator(120574 = 0 120575 = 0) 120573 = 0 Gripenberg [19] considered thenonlinear parabolic equation with Volterra integral termequation
119906119905= int
119905
0
119896 (119905 minus 119904) 120590(119906119909(119909 119904))
119909119889119904 + 119891 (119909 119905)
119909 isin (0 1) 119905 ge 0
(5)
He investigated the initial boundary value problem of (5) andestablished the global existence of a strong solution of theproblem
In the absence of the viscous term and119901-Laplace operator(120574 = 0 120575 = 0) as 120573 = 1 the model reduces to the equation
119906119905minus Δ119906 = int
119905
0
119887 (119905 minus 120591) Δ119906 (120591) 119889120591 + 119891 (119906) 119909 isin Ω 119905 gt 0
(6)
Yin [20] obtained the global existence of a classical solutionof (7) under the assumption of a one-sided growth conditionMessaoudi [21] investigated a semilinear parabolic equationwith the viscoelastic memory term He established the finitetime blow-up result for the solution with negative or vanish-ing initial energy for nonlinear function 119891(119906) = |119906|
119901minus2119906
To the best of our knowledge there are few works on thestudy of nonlinear pseudoparabolic equation with memoryterm of Volterra integral type Shang and Guo [22ndash24]investigated the initial boundary value problem and initial
value problem of the nonlinear pseudoparabolic equationswith Volterra integral term
119906119905minus 119891(119906)
119909119909minus 119906119909119909119905
+ int
119905
0
120582 (119905 minus 119904) 120590(119906 (119909 119904) 119906119909(119909 119904))
119909119889119904
= 119891 (119909 119905 119906 119906119909) 119909 isin (0 1) 119905 gt 0
(7)
They proved the existence uniqueness and regularities ofthe global strong solution and gave some conditions of thenonexistence of global solution In 2007 Ptashnyk [25] inves-tigated the initial boundary value of degenerate quasilinearpseudoparabolic equations with memory term He obtainedsome existence results of global solutions Up to now thereare not any research works on the multidimensional nonlin-ear pseudoparabolic equations with memory term
In the present work we deal with the initial boundaryproblem of the nonlinear pseudoparabolic equation with thememory term of Volterra integral type the damping termand 119901-Laplace operator
119906119905minus Δ119906 minus Δ119906
119905+ int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591
= div (|nabla119906|119901minus2nabla119906) + 1199061+120572 119909 isin Ω 119905 gt 0
119906 (119909 0) = 1199060(119909) 119909 isin Ω
119906(119909 119905)|120597Ω
= 0 119909 isin 120597Ω 119905 gt 0
(8)
whereΩ sub R119899 is a bounded domain 120582(119904) R+rarr R is a given
continuous function 120573 120574 120575 gt 0 and div(|nabla119906|119901minus2nabla119906) is theso-called 119901-Laplace operator By using the concavity methodfirst introduced by Levine [5] under negative initial energyand suitable conditions on 119901 120572 and the relaxation function120582(119905) we prove that there exists finite-time blow-up solution
Without loss of generality we choose 120573 = 120574 = 120575 = 1 inthe following discussion
2 Preliminaries and Main Results
In this section we introduce some notations basic defini-tions and important lemmas which will be needed in thispaper
For functions 119906(119909 119905) V(119909 119905) defined on Ω we introduce
(119906 V) = int
Ω
119906V 119889119909 1199062= (int
Ω
|119906|2119889119909)
12
119906119901= (int
Ω
|119906|119901119889119909)
1119901
119906119867119898 = ( sum
|120572|le119898
10038171003817100381710038171198631205721199061003817100381710038171003817
2
2)
12
119906infin= ess sup119909isinΩ
|119906 (119909)|
Δ =
1205972
1205971199092
1
+ sdot sdot sdot +
1205972
1205971199092
119899
nabla = (
120597
1205971199091
120597
120597119909119899
)
(9)
Abstract and Applied Analysis 3
We now construct a space of functions as follows Let119867119898(Ω) denote the Sobolev space with the norm 119906
119867119898 =
(sum|120572|le119898
1198631205721199062
2)
12
119862infin0(Ω) denotes the class of119862infin functions
with the compact support inΩ1198671198980(Ω) denotes the closure in
119867119898(Ω) of 119862infin
0(Ω) The Hilbert space119867119898
0(Ω) is a subspace of
the Sobolev space119867119898(Ω)The following are the basic hypotheses to establish the
main results of this paper
(a) 2 lt 119901 le 2 + 120572 lt infin
(b) 120582 is a 1198621 function satisfying
120582 (120591) ge 0 1205821015840(120591) le 0 (10)
int
infin
0
120582 (120591) 119889120591 lt
120572 (120572 + 2)
120572 (120572 + 2) + 1
(11)
To obtain the results of this paper we will introduce theldquomodifiedrdquo energy function
119864 (119905) =
1
2
(120582 ∘ nabla119906) (119905)
+
1
2
(1 minus int
119905
0
120582 (120591) 119889120591) nabla119906(119905)2
2+
1
119901
nabla119906(119905)119901
119901
minus
1
2 + 120572
int
Ω
1199062+120572
119889119909
(12)
where
(120582 ∘ V) (119905) = int
119905
0
120582 (119905 minus 120591) V(119905) minus V(120591)22119889120591 (13)
The following lemma is similar to the lemma of [21] withslight modification
Lemma 1 Assume that (10) hold Let p satisfy (119886) and let 119906 bea solution of (8) Then 119864(119905) is nonincreasing function that is
1198641015840(119905) le 0 (14)
Moreover the following energy inequality holds
119864 (119905) + int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
1198671119889120591 lt 119864 (0) (15)
Proof By multiplying the equation in (8) by 119906119905 integrating
overΩ we obtain
119889
119889119905
[int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
2119889120591 + int
119905
0
1003817100381710038171003817nabla119906119905(120591)
1003817100381710038171003817
2
2119889120591 +
1
2
int
Ω
|nabla119906|2119889119909
+
1
119901
nabla119906119901
119901minus
1
2 + 120572
int
Ω
1199062+120572
119889119909]
minus int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (120591) 119889119909 119889120591 = 0
(16)
For the last term on the left side of (16)
int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (120591) 119889119909 119889120591
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) [nabla119906 (120591) minus nabla119906 (119905)] 119889119909 119889120591
+ int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (119905) 119889119909 119889120591
= minus
1
2
int
119905
0
120582 (119905 minus 120591) [
119889
119889119905
int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909] 119889120591
+
1
2
int
119905
0
120582 (120591) [
119889
119889119905
int
Ω
|nabla119906 (119905)|2119889119909] 119889120591
= minus
1
2
119889
119889119905
[int
119905
0
120582 (119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591]
+
1
2
119889
119889119905
[int
119905
0
120582 (120591) int
Ω
|nabla119906 (119905)|2119889119909 119889120591]
+
1
2
int
119905
0
1205821015840(119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591
minus
1
2
120582 (119905) int
Ω
|nabla119906 (119905)|2119889119909
(17)
Inserting (17) into (16) we have
119889
119889119905
[int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
2119889120591 + int
119905
0
1003817100381710038171003817nabla119906119905(120591)
1003817100381710038171003817
2
2119889120591 +
1
2
int
Ω
|nabla119906|2119889119909
+
1
119901
nabla119906119901
119901minus
1
2 + 120572
int
Ω
1199062+120572
119889119909]
+
1
2
119889
119889119905
[int
119905
0
120582 (119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591]
minus
1
2
119889
119889119905
[int
119905
0
120582 (120591) int
Ω
|nabla119906 (119905)|2119889119909 119889120591]
=
1
2
int
119905
0
1205821015840(119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591
minus
1
2
120582 (119905) int
Ω
|nabla119906 (119905)|2119889119909 le 0
(18)
for regular solution The proof of Lemma 1 is completedThis result is valid for weak solutions by a simple densityargument
Nowwe consider the finite time blow-up of solutionswith119864(0) lt 0 for the problem (8)
Theorem 2 Let 119901 satisfy (a) and let the relaxation function120582(119904) be a 119862
1 function satisfying (10) and (11) Assume that1199060isin 1198671
0(Ω) such that 119864(0) lt 0 Then the solutions 119906(119909 119905)
of the problem (8) blow up in finite time that is the maximumexistence time 119879max of 119906(119909 119905) is finite and
lim119905997891rarr119879max
119906(119909 119905)2
1198671 = +infin (19)
4 Abstract and Applied Analysis
Proof The proof makes use of the so-called ldquoconcavityrdquoarguments For any 119879
0gt 0 let
119872(119905) = int
119905
0
119906(120591)2
2119889120591
+ int
119905
0
nabla119906(120591)2
2119889120591 + (119879
0minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)
(20)
for 119905 isin [0 1198790]
A direct computation yields
1198721015840(119905) = 119906(119905)
2
2+ nabla119906(119905)
2
2
minus (1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)
= 2int
119905
0
(119906 119906119905) 119889120591 + 2int
119905
0
(nabla119906 nabla119906119905) 119889120591
11987210158401015840(119905) = 2 (119906 119906
119905) + 2 (nabla119906 nabla119906
119905)
(21)
By multiplying (8) with 119906 and integrating overΩ
(119906 119906119905) + (nabla119906 nabla119906
119905)
= minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909
(22)
This implies that
11987210158401015840(119905) = minus2nabla119906 (119905)
2
2
minus 2int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus 2nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909
(23)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2
= 2119872 (119905) [minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909]
minus
120572 + 4
4
[2int
119905
0
(119906 119906119905) 119889120591 + 2int
119905
0
(nabla119906 nabla119906119905) 119889120591]
2
= 2119872 (119905) [minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906 (119905)119901
119901+ int
Ω
1199062+120572
119889119909] + (120572 + 4)
times 119867 (119905) minus [119872 (119905) minus (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)]
times [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
(24)
where
119867(119905) = [int
119905
0
(119906 119906) 119889120591 + int
119905
0
(nabla119906 nabla119906) 119889120591]
times [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
minus [int
119905
0
(119906 119906119905) 119889120591 + int
119905
0
(nabla119906 nabla119906119905) 119889120591]
2
(25)
Using Schwartzrsquos inequality we have
(int
119905
0(119906 119906119905)119889120591)
2
le int
119905
0
1199062
2119889120591int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
2119889120591
(int
119905
0(nabla119906 nabla119906
119905)119889120591)
2
le int
119905
0
nabla1199062
2119889120591int
119905
0
1003817100381710038171003817nabla119906119905
1003817100381710038171003817
2
2119889120591
int
119905
0
(119906 119906119905) 119889120591int
119905
0
(nabla119906 nabla119906119905) 119889120591
le
1
2
int
119905
0
1199062
2119889120591int
119905
0
1003817100381710038171003817nabla119906119905
1003817100381710038171003817
2
2119889120591
+
1
2
int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
2119889120591int
119905
0
nabla1199062
2119889120591
(26)
By (26) we have
119867(119905) ge 0 119905 isin [0 1198790] (27)
Thus
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120578 (119905) (28)
where
120578 (119905) = minus (120572 + 4) [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
minus 2nabla119906(119905)2
2minus 2int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus 2nabla119906(119905)119901
119901+ 2int
Ω
1199062+120572
119889119909
(29)
Abstract and Applied Analysis 5
For the third term on the left of (29) we have
minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (120591) nabla119906 (119905) 119889119909 119889120591
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) nabla [119906 (120591) minus 119906 (119905)] 119889119909 119889120591
+ int
119905
0
120582 (119905 minus 120591) nabla119906(119905)2
2119889120591
(30)
By (29) and (30) we have
120578 (119905) = minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2
+ 2int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) [nabla119906 (120591) minus nabla119906 (119905)] 119889119909 119889120591
minus 2nabla119906119901
119901+ 2int
Ω
1199062+120572
119889119909
ge minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
+ 2int
Ω
1199062+120572
119889119909
minus 2 [
120572 + 2
2
int
119905
0
120582 (119905 minus 120591) nabla119906(120591) minus nabla119906(119905)2
2119889120591
+
1
2 (120572 + 2)
int
119905
0
120582 (119905 minus 120591) nabla119906 (119905)2
2119889120591]
= minus2 (120572 + 2) 119864 (119905) + (120572 + 2) (1 minus int
119905
0
120582 (119905 minus 120591) 119889120591)
times nabla119906 (119905)2
2+
2 (120572 + 2)
119901
nabla119906119901
119901
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
= minus2 (120572 + 2) 119864 (119905) + 120572(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906 (119905)2
2
minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 +
2120572 minus 2119901 + 4
119901
nabla119906119901
119901
(31)
Using Lemma 1 we have
119864 (119905) + int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 lt 119864 (0) (32)
and then
120578 (119905) ge minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572(1 minus int
119905
0
120582 (120591) 119889120591) minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906(119905)2
2
= minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572 minus
120572 (120572 + 2) + 1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906 (119905)2
2
(33)
Since
int
infin
0
120582 (120591) 119889120591 lt
120572 (120572 + 2)
120572 (120572 + 2) + 1
(34)
this implies that
120578 (119905) gt 120575 0 le 119905 lt infin (35)
where 120575 is a positive constantFrom the discussion above we see that
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120575 (36)
From the definition of119872(119905) there exists 120588 gt 0 such that
119872(119905) ge 120588 for 119905 isin [0 119879) (37)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 120588120575 (38)
Thus
(119872(119905)minus1205724
)
10158401015840
= (minus
120572
4
)119872(119905)minus1205724minus2
[11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2]
le (minus
120572
4
) 120588120575119872(119879)minus(120572+8)4
lt 0
(39)
Hence this proves that 119872(119905)minus1205724 reaches 0 in finite time as
119905 rarr 119879minus
1 Since 119879
1is independent of 119879 we may assume that
1198791lt 119879This means
lim119905rarr119879
minus
1
119872(119905) = +infin (40)
6 Abstract and Applied Analysis
or
lim119905rarr119879
minus
1
(int
119905
0
119906(120591)2
2119889120591 + int
119905
0
nabla119906(120591)2
2119889120591
+ (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)) = +infin
(41)
This implies that
lim119905rarr119879
minus
1
int
119905
0
119906(120591)2
1198671
119889120591 = +infin (42)
Then the desired assertion immediately follows
Remark 1 In the absence of the viscous term (Δ119906119905) or 119901-
Laplace operator (div(|nabla119906|119901minus2nabla119906)) for the problem (8) theequation reduced to
(1) 119906119905minus Δ119906 + int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(2) 119906119905minus Δ119906 minus Δ119906
119905+ int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(3) 119906119905minusΔ119906+int
119905
0120582(119905minus120591)Δ119906(120591)119889120591 = div(|nabla119906|119901minus2nabla119906)+1199061+120572
and from the process of the proof we can see that the resultsof Theorem 2 still hold
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the NSF of China (40890150 and40890153) and the Scientific Program (2008B080701042) ofGuangdong Province The authors thank the anonymousreferees for their helpful comments and suggestions
References
[1] A B AlrsquoshinMO Korpusov andA G Sveshnikov SveshnikovBlow-up in Nonlinear Sobolev Type Equations De Gruyter 2011
[2] A Y Kolesov E F Mishchenko and N K Rozov ldquoAsymptoticmethods for studying periodic solutions of nonlinear hyper-bolic equationsrdquo Trudy Matematicheskogo Instituta Imeni V ASteklova vol 222 pp 3ndash198 1998
[3] BK Shivamoggi ldquoA symmetric regularized long-wave equationfor shallow water wavesrdquoThe Physics of Fluids vol 29 no 3 pp890ndash891 1986
[4] P Rosenau ldquoEvolution and breaking of the ion-acoustic wavesrdquoPhysics of Fluids vol 31 no 6 pp 1317ndash1319 1988
[5] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119891 (119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[6] M X Wang and X Q Ding ldquoGlobal solutions asymptoticbehavior and blow-up problems for semilinear heat equationsrdquoScience in China A vol 35 pp 1026ndash1034 1992
[7] F Gazzola and T Weth ldquoFinite time blow-up and globalsolutions for semilinear parabolic equations with initial data athigh energy levelrdquo Differential and Integral Equations vol 18no 9 pp 961ndash990 2005
[8] Y C Liu R Z Xu and T Yu ldquoGlobal existence nonexistenceand asymptotic behavior of solutions for the Cauchy problem ofsemilinear heat equationsrdquoNonlinear AnalysisTheory Methodsamp Applications vol 68 no 11 pp 3332ndash3348 2008
[9] H Wang and Y He ldquoOn blow-up of solutions for a semilinearparabolic equation involving variable source and positive initialenergyrdquo Applied Mathematics Letters vol 26 no 10 pp 1008ndash1012 2013
[10] E I Kaikina P I Naumkin and I Shishmarev ldquoA periodicproblem for a Sobolev-type nonlinear equationrdquo FunctionAnalysis and Its Applications vol 44 no 3 pp 14ndash26 2010
[11] Y Cao J X Yin and C H Jin ldquoA periodic problem of asemilinear pseudoparabolic equationrdquo Abstract and AppliedAnalysis vol 2011 Article ID 363579 27 pages 2011
[12] Y Cao J Yin and C Wang ldquoCauchy problems of semilinearpseudo-parabolic equationsrdquo Journal of Differential Equationsvol 246 no 12 pp 4568ndash4590 2009
[13] E I Kaikina ldquoInitial boundary value problems for nonlinearpseudo-parabolic equations in a critical caserdquo Electronic Journalof Differential Equation vol 109 pp 1ndash25 2007
[14] R Xu and J Su ldquoGlobal existence and finite time blow-up fora class of semilinear pseudo-parabolic equationsrdquo Journal ofFunctional Analysis vol 264 no 12 pp 2732ndash2763 2013
[15] M Tsutsumi ldquoExistence and nonexistence of global solutionsfor nonlinear parabolic equationsrdquo Publications of the ResearchInstitute for Mathematical Sciences vol 8 no 2 pp 211ndash2291972
[16] Y C Liu and J S Zhao ldquoNonlinear parabolic equations withcritical initial conditions 119869
(1199060)= 119889 or 119868
(1199060)= 0rdquo Nonlinear
Analysis Theory Methods amp Applications vol 58 no 7-8 pp873ndash883 2004
[17] R Xu X Cao and T Yu ldquoFinite time blow-up and globalsolutions for a class of semilinear parabolic equations at highenergy levelrdquo Nonlinear Analysis Real World Applications vol13 no 1 pp 197ndash202 2012
[18] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[19] G Gripenberg ldquoGlobal existence of solutions of Volterraintegrodifferential equations of parabolic typerdquo Journal of Dif-ferential Equations vol 102 no 2 pp 382ndash390 1993
[20] H Yin ldquoWeak and classical solutions of some nonlinear volterraintegrodifferential equationsrdquo Communications in Partial Dif-ferential Equations vol 17 no 7-8 pp 1369ndash1385 1992
[21] S A Messaoudi ldquoBlow-up of solutions of a semi linear heatequation with a Visco-elastic termrdquo in Nonlinear Elliptic andParabolic Problems H Brezis M Chipot and J Escher Edsvol 64 of Progress in Nonlinear Differential Equations andTheirApplications pp 351ndash356 Birkhauser Basel Switzerland 2005
[22] Y D Shang and B L Guo ldquoOn the problem of the existenceof global solutions for a class of nonlinear convolutionalintegro-differential equations of pseudoparabolic typerdquo ActaMathematicaeApplicatae Sinica vol 26 no 3 pp 512ndash524 2003
[23] Y D Shang and B L Guo ldquoInitial boundary value problemand initial value problem for the nonlinear integro-differentialequations of pseudoparabolic typerdquoMathematicaApplicata vol15 no 1 pp 40ndash45 2002
Abstract and Applied Analysis 7
[24] Y D Shang and B L Guo ldquoInitial-boundary value problemsfor a class of quasilinear integro-differential equations of pseu-doparabolic typerdquo Chinese Journal of Engineering Mathematicsvol 20 no 4 pp 1ndash6 2003
[25] M Ptashnyk ldquoDegenerate quaslinear pseudoparabolic equa-tions with memory terms and variational inequalitiesrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 66 no 12 pp2653ndash2675 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
solutions were proved and the large-time behavior wasinvestigated Kaikina [13] and Xu and Su [14] discussedthe initial boundary value problems of pseudoparabolicequation (3) under some classes of nonlinear function 119891(119906)They obtained some sufficient conditions of existence anduniqueness of local solutions and the large-time behavior ofglobal solutions
In the absence of the memory term and the viscous term(120582(119904) = 0 120574 = 0) 120573 = 0 120575 = 1 (1) becomes nonlinearparabolic equation with 119901-Laplace nonlinear term
119906119905= div (|nabla119906|119901minus2nabla119906) + 119891 (119906) 119909 isin Ω 119905 gt 0 (4)
Tsutsumi [15] studied the initial boundary value problem of(4) with 119891(119906) = 119906
1+120572 where 119901 lt 2 + 120572 He obtained theexistence of global weak solutions by using the potential wellmethod Liu and Zhao [16] considered the same problemwith critical initial conditions 119869(119906
0) = 119889 or 119868(119906
0) = 0 and
proved the existence of global solution for this problem Xuet al [17] discussed (4) at the high energy level where 119901 lt
2 + 120572 lt infin if 119899 le 119901 and 119901 lt 2 + 120572 le 119899119901(119899 minus 119901) if119899 gt 119901They proved the finite time blow-up of solutions by thecomparison principle and variational methods Messaoudi in[18] considers an initial boundary value problem related to (4)and proves under suitable conditions on 119891 a blow-up resultfor solutions with vanishing or negative initial energy
In the absence of the viscous term and119901-Laplace operator(120574 = 0 120575 = 0) 120573 = 0 Gripenberg [19] considered thenonlinear parabolic equation with Volterra integral termequation
119906119905= int
119905
0
119896 (119905 minus 119904) 120590(119906119909(119909 119904))
119909119889119904 + 119891 (119909 119905)
119909 isin (0 1) 119905 ge 0
(5)
He investigated the initial boundary value problem of (5) andestablished the global existence of a strong solution of theproblem
In the absence of the viscous term and119901-Laplace operator(120574 = 0 120575 = 0) as 120573 = 1 the model reduces to the equation
119906119905minus Δ119906 = int
119905
0
119887 (119905 minus 120591) Δ119906 (120591) 119889120591 + 119891 (119906) 119909 isin Ω 119905 gt 0
(6)
Yin [20] obtained the global existence of a classical solutionof (7) under the assumption of a one-sided growth conditionMessaoudi [21] investigated a semilinear parabolic equationwith the viscoelastic memory term He established the finitetime blow-up result for the solution with negative or vanish-ing initial energy for nonlinear function 119891(119906) = |119906|
119901minus2119906
To the best of our knowledge there are few works on thestudy of nonlinear pseudoparabolic equation with memoryterm of Volterra integral type Shang and Guo [22ndash24]investigated the initial boundary value problem and initial
value problem of the nonlinear pseudoparabolic equationswith Volterra integral term
119906119905minus 119891(119906)
119909119909minus 119906119909119909119905
+ int
119905
0
120582 (119905 minus 119904) 120590(119906 (119909 119904) 119906119909(119909 119904))
119909119889119904
= 119891 (119909 119905 119906 119906119909) 119909 isin (0 1) 119905 gt 0
(7)
They proved the existence uniqueness and regularities ofthe global strong solution and gave some conditions of thenonexistence of global solution In 2007 Ptashnyk [25] inves-tigated the initial boundary value of degenerate quasilinearpseudoparabolic equations with memory term He obtainedsome existence results of global solutions Up to now thereare not any research works on the multidimensional nonlin-ear pseudoparabolic equations with memory term
In the present work we deal with the initial boundaryproblem of the nonlinear pseudoparabolic equation with thememory term of Volterra integral type the damping termand 119901-Laplace operator
119906119905minus Δ119906 minus Δ119906
119905+ int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591
= div (|nabla119906|119901minus2nabla119906) + 1199061+120572 119909 isin Ω 119905 gt 0
119906 (119909 0) = 1199060(119909) 119909 isin Ω
119906(119909 119905)|120597Ω
= 0 119909 isin 120597Ω 119905 gt 0
(8)
whereΩ sub R119899 is a bounded domain 120582(119904) R+rarr R is a given
continuous function 120573 120574 120575 gt 0 and div(|nabla119906|119901minus2nabla119906) is theso-called 119901-Laplace operator By using the concavity methodfirst introduced by Levine [5] under negative initial energyand suitable conditions on 119901 120572 and the relaxation function120582(119905) we prove that there exists finite-time blow-up solution
Without loss of generality we choose 120573 = 120574 = 120575 = 1 inthe following discussion
2 Preliminaries and Main Results
In this section we introduce some notations basic defini-tions and important lemmas which will be needed in thispaper
For functions 119906(119909 119905) V(119909 119905) defined on Ω we introduce
(119906 V) = int
Ω
119906V 119889119909 1199062= (int
Ω
|119906|2119889119909)
12
119906119901= (int
Ω
|119906|119901119889119909)
1119901
119906119867119898 = ( sum
|120572|le119898
10038171003817100381710038171198631205721199061003817100381710038171003817
2
2)
12
119906infin= ess sup119909isinΩ
|119906 (119909)|
Δ =
1205972
1205971199092
1
+ sdot sdot sdot +
1205972
1205971199092
119899
nabla = (
120597
1205971199091
120597
120597119909119899
)
(9)
Abstract and Applied Analysis 3
We now construct a space of functions as follows Let119867119898(Ω) denote the Sobolev space with the norm 119906
119867119898 =
(sum|120572|le119898
1198631205721199062
2)
12
119862infin0(Ω) denotes the class of119862infin functions
with the compact support inΩ1198671198980(Ω) denotes the closure in
119867119898(Ω) of 119862infin
0(Ω) The Hilbert space119867119898
0(Ω) is a subspace of
the Sobolev space119867119898(Ω)The following are the basic hypotheses to establish the
main results of this paper
(a) 2 lt 119901 le 2 + 120572 lt infin
(b) 120582 is a 1198621 function satisfying
120582 (120591) ge 0 1205821015840(120591) le 0 (10)
int
infin
0
120582 (120591) 119889120591 lt
120572 (120572 + 2)
120572 (120572 + 2) + 1
(11)
To obtain the results of this paper we will introduce theldquomodifiedrdquo energy function
119864 (119905) =
1
2
(120582 ∘ nabla119906) (119905)
+
1
2
(1 minus int
119905
0
120582 (120591) 119889120591) nabla119906(119905)2
2+
1
119901
nabla119906(119905)119901
119901
minus
1
2 + 120572
int
Ω
1199062+120572
119889119909
(12)
where
(120582 ∘ V) (119905) = int
119905
0
120582 (119905 minus 120591) V(119905) minus V(120591)22119889120591 (13)
The following lemma is similar to the lemma of [21] withslight modification
Lemma 1 Assume that (10) hold Let p satisfy (119886) and let 119906 bea solution of (8) Then 119864(119905) is nonincreasing function that is
1198641015840(119905) le 0 (14)
Moreover the following energy inequality holds
119864 (119905) + int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
1198671119889120591 lt 119864 (0) (15)
Proof By multiplying the equation in (8) by 119906119905 integrating
overΩ we obtain
119889
119889119905
[int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
2119889120591 + int
119905
0
1003817100381710038171003817nabla119906119905(120591)
1003817100381710038171003817
2
2119889120591 +
1
2
int
Ω
|nabla119906|2119889119909
+
1
119901
nabla119906119901
119901minus
1
2 + 120572
int
Ω
1199062+120572
119889119909]
minus int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (120591) 119889119909 119889120591 = 0
(16)
For the last term on the left side of (16)
int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (120591) 119889119909 119889120591
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) [nabla119906 (120591) minus nabla119906 (119905)] 119889119909 119889120591
+ int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (119905) 119889119909 119889120591
= minus
1
2
int
119905
0
120582 (119905 minus 120591) [
119889
119889119905
int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909] 119889120591
+
1
2
int
119905
0
120582 (120591) [
119889
119889119905
int
Ω
|nabla119906 (119905)|2119889119909] 119889120591
= minus
1
2
119889
119889119905
[int
119905
0
120582 (119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591]
+
1
2
119889
119889119905
[int
119905
0
120582 (120591) int
Ω
|nabla119906 (119905)|2119889119909 119889120591]
+
1
2
int
119905
0
1205821015840(119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591
minus
1
2
120582 (119905) int
Ω
|nabla119906 (119905)|2119889119909
(17)
Inserting (17) into (16) we have
119889
119889119905
[int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
2119889120591 + int
119905
0
1003817100381710038171003817nabla119906119905(120591)
1003817100381710038171003817
2
2119889120591 +
1
2
int
Ω
|nabla119906|2119889119909
+
1
119901
nabla119906119901
119901minus
1
2 + 120572
int
Ω
1199062+120572
119889119909]
+
1
2
119889
119889119905
[int
119905
0
120582 (119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591]
minus
1
2
119889
119889119905
[int
119905
0
120582 (120591) int
Ω
|nabla119906 (119905)|2119889119909 119889120591]
=
1
2
int
119905
0
1205821015840(119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591
minus
1
2
120582 (119905) int
Ω
|nabla119906 (119905)|2119889119909 le 0
(18)
for regular solution The proof of Lemma 1 is completedThis result is valid for weak solutions by a simple densityargument
Nowwe consider the finite time blow-up of solutionswith119864(0) lt 0 for the problem (8)
Theorem 2 Let 119901 satisfy (a) and let the relaxation function120582(119904) be a 119862
1 function satisfying (10) and (11) Assume that1199060isin 1198671
0(Ω) such that 119864(0) lt 0 Then the solutions 119906(119909 119905)
of the problem (8) blow up in finite time that is the maximumexistence time 119879max of 119906(119909 119905) is finite and
lim119905997891rarr119879max
119906(119909 119905)2
1198671 = +infin (19)
4 Abstract and Applied Analysis
Proof The proof makes use of the so-called ldquoconcavityrdquoarguments For any 119879
0gt 0 let
119872(119905) = int
119905
0
119906(120591)2
2119889120591
+ int
119905
0
nabla119906(120591)2
2119889120591 + (119879
0minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)
(20)
for 119905 isin [0 1198790]
A direct computation yields
1198721015840(119905) = 119906(119905)
2
2+ nabla119906(119905)
2
2
minus (1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)
= 2int
119905
0
(119906 119906119905) 119889120591 + 2int
119905
0
(nabla119906 nabla119906119905) 119889120591
11987210158401015840(119905) = 2 (119906 119906
119905) + 2 (nabla119906 nabla119906
119905)
(21)
By multiplying (8) with 119906 and integrating overΩ
(119906 119906119905) + (nabla119906 nabla119906
119905)
= minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909
(22)
This implies that
11987210158401015840(119905) = minus2nabla119906 (119905)
2
2
minus 2int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus 2nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909
(23)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2
= 2119872 (119905) [minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909]
minus
120572 + 4
4
[2int
119905
0
(119906 119906119905) 119889120591 + 2int
119905
0
(nabla119906 nabla119906119905) 119889120591]
2
= 2119872 (119905) [minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906 (119905)119901
119901+ int
Ω
1199062+120572
119889119909] + (120572 + 4)
times 119867 (119905) minus [119872 (119905) minus (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)]
times [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
(24)
where
119867(119905) = [int
119905
0
(119906 119906) 119889120591 + int
119905
0
(nabla119906 nabla119906) 119889120591]
times [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
minus [int
119905
0
(119906 119906119905) 119889120591 + int
119905
0
(nabla119906 nabla119906119905) 119889120591]
2
(25)
Using Schwartzrsquos inequality we have
(int
119905
0(119906 119906119905)119889120591)
2
le int
119905
0
1199062
2119889120591int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
2119889120591
(int
119905
0(nabla119906 nabla119906
119905)119889120591)
2
le int
119905
0
nabla1199062
2119889120591int
119905
0
1003817100381710038171003817nabla119906119905
1003817100381710038171003817
2
2119889120591
int
119905
0
(119906 119906119905) 119889120591int
119905
0
(nabla119906 nabla119906119905) 119889120591
le
1
2
int
119905
0
1199062
2119889120591int
119905
0
1003817100381710038171003817nabla119906119905
1003817100381710038171003817
2
2119889120591
+
1
2
int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
2119889120591int
119905
0
nabla1199062
2119889120591
(26)
By (26) we have
119867(119905) ge 0 119905 isin [0 1198790] (27)
Thus
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120578 (119905) (28)
where
120578 (119905) = minus (120572 + 4) [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
minus 2nabla119906(119905)2
2minus 2int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus 2nabla119906(119905)119901
119901+ 2int
Ω
1199062+120572
119889119909
(29)
Abstract and Applied Analysis 5
For the third term on the left of (29) we have
minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (120591) nabla119906 (119905) 119889119909 119889120591
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) nabla [119906 (120591) minus 119906 (119905)] 119889119909 119889120591
+ int
119905
0
120582 (119905 minus 120591) nabla119906(119905)2
2119889120591
(30)
By (29) and (30) we have
120578 (119905) = minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2
+ 2int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) [nabla119906 (120591) minus nabla119906 (119905)] 119889119909 119889120591
minus 2nabla119906119901
119901+ 2int
Ω
1199062+120572
119889119909
ge minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
+ 2int
Ω
1199062+120572
119889119909
minus 2 [
120572 + 2
2
int
119905
0
120582 (119905 minus 120591) nabla119906(120591) minus nabla119906(119905)2
2119889120591
+
1
2 (120572 + 2)
int
119905
0
120582 (119905 minus 120591) nabla119906 (119905)2
2119889120591]
= minus2 (120572 + 2) 119864 (119905) + (120572 + 2) (1 minus int
119905
0
120582 (119905 minus 120591) 119889120591)
times nabla119906 (119905)2
2+
2 (120572 + 2)
119901
nabla119906119901
119901
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
= minus2 (120572 + 2) 119864 (119905) + 120572(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906 (119905)2
2
minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 +
2120572 minus 2119901 + 4
119901
nabla119906119901
119901
(31)
Using Lemma 1 we have
119864 (119905) + int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 lt 119864 (0) (32)
and then
120578 (119905) ge minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572(1 minus int
119905
0
120582 (120591) 119889120591) minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906(119905)2
2
= minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572 minus
120572 (120572 + 2) + 1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906 (119905)2
2
(33)
Since
int
infin
0
120582 (120591) 119889120591 lt
120572 (120572 + 2)
120572 (120572 + 2) + 1
(34)
this implies that
120578 (119905) gt 120575 0 le 119905 lt infin (35)
where 120575 is a positive constantFrom the discussion above we see that
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120575 (36)
From the definition of119872(119905) there exists 120588 gt 0 such that
119872(119905) ge 120588 for 119905 isin [0 119879) (37)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 120588120575 (38)
Thus
(119872(119905)minus1205724
)
10158401015840
= (minus
120572
4
)119872(119905)minus1205724minus2
[11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2]
le (minus
120572
4
) 120588120575119872(119879)minus(120572+8)4
lt 0
(39)
Hence this proves that 119872(119905)minus1205724 reaches 0 in finite time as
119905 rarr 119879minus
1 Since 119879
1is independent of 119879 we may assume that
1198791lt 119879This means
lim119905rarr119879
minus
1
119872(119905) = +infin (40)
6 Abstract and Applied Analysis
or
lim119905rarr119879
minus
1
(int
119905
0
119906(120591)2
2119889120591 + int
119905
0
nabla119906(120591)2
2119889120591
+ (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)) = +infin
(41)
This implies that
lim119905rarr119879
minus
1
int
119905
0
119906(120591)2
1198671
119889120591 = +infin (42)
Then the desired assertion immediately follows
Remark 1 In the absence of the viscous term (Δ119906119905) or 119901-
Laplace operator (div(|nabla119906|119901minus2nabla119906)) for the problem (8) theequation reduced to
(1) 119906119905minus Δ119906 + int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(2) 119906119905minus Δ119906 minus Δ119906
119905+ int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(3) 119906119905minusΔ119906+int
119905
0120582(119905minus120591)Δ119906(120591)119889120591 = div(|nabla119906|119901minus2nabla119906)+1199061+120572
and from the process of the proof we can see that the resultsof Theorem 2 still hold
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the NSF of China (40890150 and40890153) and the Scientific Program (2008B080701042) ofGuangdong Province The authors thank the anonymousreferees for their helpful comments and suggestions
References
[1] A B AlrsquoshinMO Korpusov andA G Sveshnikov SveshnikovBlow-up in Nonlinear Sobolev Type Equations De Gruyter 2011
[2] A Y Kolesov E F Mishchenko and N K Rozov ldquoAsymptoticmethods for studying periodic solutions of nonlinear hyper-bolic equationsrdquo Trudy Matematicheskogo Instituta Imeni V ASteklova vol 222 pp 3ndash198 1998
[3] BK Shivamoggi ldquoA symmetric regularized long-wave equationfor shallow water wavesrdquoThe Physics of Fluids vol 29 no 3 pp890ndash891 1986
[4] P Rosenau ldquoEvolution and breaking of the ion-acoustic wavesrdquoPhysics of Fluids vol 31 no 6 pp 1317ndash1319 1988
[5] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119891 (119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[6] M X Wang and X Q Ding ldquoGlobal solutions asymptoticbehavior and blow-up problems for semilinear heat equationsrdquoScience in China A vol 35 pp 1026ndash1034 1992
[7] F Gazzola and T Weth ldquoFinite time blow-up and globalsolutions for semilinear parabolic equations with initial data athigh energy levelrdquo Differential and Integral Equations vol 18no 9 pp 961ndash990 2005
[8] Y C Liu R Z Xu and T Yu ldquoGlobal existence nonexistenceand asymptotic behavior of solutions for the Cauchy problem ofsemilinear heat equationsrdquoNonlinear AnalysisTheory Methodsamp Applications vol 68 no 11 pp 3332ndash3348 2008
[9] H Wang and Y He ldquoOn blow-up of solutions for a semilinearparabolic equation involving variable source and positive initialenergyrdquo Applied Mathematics Letters vol 26 no 10 pp 1008ndash1012 2013
[10] E I Kaikina P I Naumkin and I Shishmarev ldquoA periodicproblem for a Sobolev-type nonlinear equationrdquo FunctionAnalysis and Its Applications vol 44 no 3 pp 14ndash26 2010
[11] Y Cao J X Yin and C H Jin ldquoA periodic problem of asemilinear pseudoparabolic equationrdquo Abstract and AppliedAnalysis vol 2011 Article ID 363579 27 pages 2011
[12] Y Cao J Yin and C Wang ldquoCauchy problems of semilinearpseudo-parabolic equationsrdquo Journal of Differential Equationsvol 246 no 12 pp 4568ndash4590 2009
[13] E I Kaikina ldquoInitial boundary value problems for nonlinearpseudo-parabolic equations in a critical caserdquo Electronic Journalof Differential Equation vol 109 pp 1ndash25 2007
[14] R Xu and J Su ldquoGlobal existence and finite time blow-up fora class of semilinear pseudo-parabolic equationsrdquo Journal ofFunctional Analysis vol 264 no 12 pp 2732ndash2763 2013
[15] M Tsutsumi ldquoExistence and nonexistence of global solutionsfor nonlinear parabolic equationsrdquo Publications of the ResearchInstitute for Mathematical Sciences vol 8 no 2 pp 211ndash2291972
[16] Y C Liu and J S Zhao ldquoNonlinear parabolic equations withcritical initial conditions 119869
(1199060)= 119889 or 119868
(1199060)= 0rdquo Nonlinear
Analysis Theory Methods amp Applications vol 58 no 7-8 pp873ndash883 2004
[17] R Xu X Cao and T Yu ldquoFinite time blow-up and globalsolutions for a class of semilinear parabolic equations at highenergy levelrdquo Nonlinear Analysis Real World Applications vol13 no 1 pp 197ndash202 2012
[18] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[19] G Gripenberg ldquoGlobal existence of solutions of Volterraintegrodifferential equations of parabolic typerdquo Journal of Dif-ferential Equations vol 102 no 2 pp 382ndash390 1993
[20] H Yin ldquoWeak and classical solutions of some nonlinear volterraintegrodifferential equationsrdquo Communications in Partial Dif-ferential Equations vol 17 no 7-8 pp 1369ndash1385 1992
[21] S A Messaoudi ldquoBlow-up of solutions of a semi linear heatequation with a Visco-elastic termrdquo in Nonlinear Elliptic andParabolic Problems H Brezis M Chipot and J Escher Edsvol 64 of Progress in Nonlinear Differential Equations andTheirApplications pp 351ndash356 Birkhauser Basel Switzerland 2005
[22] Y D Shang and B L Guo ldquoOn the problem of the existenceof global solutions for a class of nonlinear convolutionalintegro-differential equations of pseudoparabolic typerdquo ActaMathematicaeApplicatae Sinica vol 26 no 3 pp 512ndash524 2003
[23] Y D Shang and B L Guo ldquoInitial boundary value problemand initial value problem for the nonlinear integro-differentialequations of pseudoparabolic typerdquoMathematicaApplicata vol15 no 1 pp 40ndash45 2002
Abstract and Applied Analysis 7
[24] Y D Shang and B L Guo ldquoInitial-boundary value problemsfor a class of quasilinear integro-differential equations of pseu-doparabolic typerdquo Chinese Journal of Engineering Mathematicsvol 20 no 4 pp 1ndash6 2003
[25] M Ptashnyk ldquoDegenerate quaslinear pseudoparabolic equa-tions with memory terms and variational inequalitiesrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 66 no 12 pp2653ndash2675 2007
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
We now construct a space of functions as follows Let119867119898(Ω) denote the Sobolev space with the norm 119906
119867119898 =
(sum|120572|le119898
1198631205721199062
2)
12
119862infin0(Ω) denotes the class of119862infin functions
with the compact support inΩ1198671198980(Ω) denotes the closure in
119867119898(Ω) of 119862infin
0(Ω) The Hilbert space119867119898
0(Ω) is a subspace of
the Sobolev space119867119898(Ω)The following are the basic hypotheses to establish the
main results of this paper
(a) 2 lt 119901 le 2 + 120572 lt infin
(b) 120582 is a 1198621 function satisfying
120582 (120591) ge 0 1205821015840(120591) le 0 (10)
int
infin
0
120582 (120591) 119889120591 lt
120572 (120572 + 2)
120572 (120572 + 2) + 1
(11)
To obtain the results of this paper we will introduce theldquomodifiedrdquo energy function
119864 (119905) =
1
2
(120582 ∘ nabla119906) (119905)
+
1
2
(1 minus int
119905
0
120582 (120591) 119889120591) nabla119906(119905)2
2+
1
119901
nabla119906(119905)119901
119901
minus
1
2 + 120572
int
Ω
1199062+120572
119889119909
(12)
where
(120582 ∘ V) (119905) = int
119905
0
120582 (119905 minus 120591) V(119905) minus V(120591)22119889120591 (13)
The following lemma is similar to the lemma of [21] withslight modification
Lemma 1 Assume that (10) hold Let p satisfy (119886) and let 119906 bea solution of (8) Then 119864(119905) is nonincreasing function that is
1198641015840(119905) le 0 (14)
Moreover the following energy inequality holds
119864 (119905) + int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
1198671119889120591 lt 119864 (0) (15)
Proof By multiplying the equation in (8) by 119906119905 integrating
overΩ we obtain
119889
119889119905
[int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
2119889120591 + int
119905
0
1003817100381710038171003817nabla119906119905(120591)
1003817100381710038171003817
2
2119889120591 +
1
2
int
Ω
|nabla119906|2119889119909
+
1
119901
nabla119906119901
119901minus
1
2 + 120572
int
Ω
1199062+120572
119889119909]
minus int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (120591) 119889119909 119889120591 = 0
(16)
For the last term on the left side of (16)
int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (120591) 119889119909 119889120591
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) [nabla119906 (120591) minus nabla119906 (119905)] 119889119909 119889120591
+ int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906119905(119905) nabla119906 (119905) 119889119909 119889120591
= minus
1
2
int
119905
0
120582 (119905 minus 120591) [
119889
119889119905
int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909] 119889120591
+
1
2
int
119905
0
120582 (120591) [
119889
119889119905
int
Ω
|nabla119906 (119905)|2119889119909] 119889120591
= minus
1
2
119889
119889119905
[int
119905
0
120582 (119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591]
+
1
2
119889
119889119905
[int
119905
0
120582 (120591) int
Ω
|nabla119906 (119905)|2119889119909 119889120591]
+
1
2
int
119905
0
1205821015840(119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591
minus
1
2
120582 (119905) int
Ω
|nabla119906 (119905)|2119889119909
(17)
Inserting (17) into (16) we have
119889
119889119905
[int
119905
0
1003817100381710038171003817119906119905(120591)
1003817100381710038171003817
2
2119889120591 + int
119905
0
1003817100381710038171003817nabla119906119905(120591)
1003817100381710038171003817
2
2119889120591 +
1
2
int
Ω
|nabla119906|2119889119909
+
1
119901
nabla119906119901
119901minus
1
2 + 120572
int
Ω
1199062+120572
119889119909]
+
1
2
119889
119889119905
[int
119905
0
120582 (119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591]
minus
1
2
119889
119889119905
[int
119905
0
120582 (120591) int
Ω
|nabla119906 (119905)|2119889119909 119889120591]
=
1
2
int
119905
0
1205821015840(119905 minus 120591) int
Ω
|nabla119906 (120591) minus nabla119906 (119905)|2119889119909 119889120591
minus
1
2
120582 (119905) int
Ω
|nabla119906 (119905)|2119889119909 le 0
(18)
for regular solution The proof of Lemma 1 is completedThis result is valid for weak solutions by a simple densityargument
Nowwe consider the finite time blow-up of solutionswith119864(0) lt 0 for the problem (8)
Theorem 2 Let 119901 satisfy (a) and let the relaxation function120582(119904) be a 119862
1 function satisfying (10) and (11) Assume that1199060isin 1198671
0(Ω) such that 119864(0) lt 0 Then the solutions 119906(119909 119905)
of the problem (8) blow up in finite time that is the maximumexistence time 119879max of 119906(119909 119905) is finite and
lim119905997891rarr119879max
119906(119909 119905)2
1198671 = +infin (19)
4 Abstract and Applied Analysis
Proof The proof makes use of the so-called ldquoconcavityrdquoarguments For any 119879
0gt 0 let
119872(119905) = int
119905
0
119906(120591)2
2119889120591
+ int
119905
0
nabla119906(120591)2
2119889120591 + (119879
0minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)
(20)
for 119905 isin [0 1198790]
A direct computation yields
1198721015840(119905) = 119906(119905)
2
2+ nabla119906(119905)
2
2
minus (1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)
= 2int
119905
0
(119906 119906119905) 119889120591 + 2int
119905
0
(nabla119906 nabla119906119905) 119889120591
11987210158401015840(119905) = 2 (119906 119906
119905) + 2 (nabla119906 nabla119906
119905)
(21)
By multiplying (8) with 119906 and integrating overΩ
(119906 119906119905) + (nabla119906 nabla119906
119905)
= minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909
(22)
This implies that
11987210158401015840(119905) = minus2nabla119906 (119905)
2
2
minus 2int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus 2nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909
(23)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2
= 2119872 (119905) [minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909]
minus
120572 + 4
4
[2int
119905
0
(119906 119906119905) 119889120591 + 2int
119905
0
(nabla119906 nabla119906119905) 119889120591]
2
= 2119872 (119905) [minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906 (119905)119901
119901+ int
Ω
1199062+120572
119889119909] + (120572 + 4)
times 119867 (119905) minus [119872 (119905) minus (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)]
times [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
(24)
where
119867(119905) = [int
119905
0
(119906 119906) 119889120591 + int
119905
0
(nabla119906 nabla119906) 119889120591]
times [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
minus [int
119905
0
(119906 119906119905) 119889120591 + int
119905
0
(nabla119906 nabla119906119905) 119889120591]
2
(25)
Using Schwartzrsquos inequality we have
(int
119905
0(119906 119906119905)119889120591)
2
le int
119905
0
1199062
2119889120591int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
2119889120591
(int
119905
0(nabla119906 nabla119906
119905)119889120591)
2
le int
119905
0
nabla1199062
2119889120591int
119905
0
1003817100381710038171003817nabla119906119905
1003817100381710038171003817
2
2119889120591
int
119905
0
(119906 119906119905) 119889120591int
119905
0
(nabla119906 nabla119906119905) 119889120591
le
1
2
int
119905
0
1199062
2119889120591int
119905
0
1003817100381710038171003817nabla119906119905
1003817100381710038171003817
2
2119889120591
+
1
2
int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
2119889120591int
119905
0
nabla1199062
2119889120591
(26)
By (26) we have
119867(119905) ge 0 119905 isin [0 1198790] (27)
Thus
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120578 (119905) (28)
where
120578 (119905) = minus (120572 + 4) [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
minus 2nabla119906(119905)2
2minus 2int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus 2nabla119906(119905)119901
119901+ 2int
Ω
1199062+120572
119889119909
(29)
Abstract and Applied Analysis 5
For the third term on the left of (29) we have
minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (120591) nabla119906 (119905) 119889119909 119889120591
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) nabla [119906 (120591) minus 119906 (119905)] 119889119909 119889120591
+ int
119905
0
120582 (119905 minus 120591) nabla119906(119905)2
2119889120591
(30)
By (29) and (30) we have
120578 (119905) = minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2
+ 2int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) [nabla119906 (120591) minus nabla119906 (119905)] 119889119909 119889120591
minus 2nabla119906119901
119901+ 2int
Ω
1199062+120572
119889119909
ge minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
+ 2int
Ω
1199062+120572
119889119909
minus 2 [
120572 + 2
2
int
119905
0
120582 (119905 minus 120591) nabla119906(120591) minus nabla119906(119905)2
2119889120591
+
1
2 (120572 + 2)
int
119905
0
120582 (119905 minus 120591) nabla119906 (119905)2
2119889120591]
= minus2 (120572 + 2) 119864 (119905) + (120572 + 2) (1 minus int
119905
0
120582 (119905 minus 120591) 119889120591)
times nabla119906 (119905)2
2+
2 (120572 + 2)
119901
nabla119906119901
119901
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
= minus2 (120572 + 2) 119864 (119905) + 120572(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906 (119905)2
2
minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 +
2120572 minus 2119901 + 4
119901
nabla119906119901
119901
(31)
Using Lemma 1 we have
119864 (119905) + int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 lt 119864 (0) (32)
and then
120578 (119905) ge minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572(1 minus int
119905
0
120582 (120591) 119889120591) minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906(119905)2
2
= minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572 minus
120572 (120572 + 2) + 1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906 (119905)2
2
(33)
Since
int
infin
0
120582 (120591) 119889120591 lt
120572 (120572 + 2)
120572 (120572 + 2) + 1
(34)
this implies that
120578 (119905) gt 120575 0 le 119905 lt infin (35)
where 120575 is a positive constantFrom the discussion above we see that
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120575 (36)
From the definition of119872(119905) there exists 120588 gt 0 such that
119872(119905) ge 120588 for 119905 isin [0 119879) (37)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 120588120575 (38)
Thus
(119872(119905)minus1205724
)
10158401015840
= (minus
120572
4
)119872(119905)minus1205724minus2
[11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2]
le (minus
120572
4
) 120588120575119872(119879)minus(120572+8)4
lt 0
(39)
Hence this proves that 119872(119905)minus1205724 reaches 0 in finite time as
119905 rarr 119879minus
1 Since 119879
1is independent of 119879 we may assume that
1198791lt 119879This means
lim119905rarr119879
minus
1
119872(119905) = +infin (40)
6 Abstract and Applied Analysis
or
lim119905rarr119879
minus
1
(int
119905
0
119906(120591)2
2119889120591 + int
119905
0
nabla119906(120591)2
2119889120591
+ (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)) = +infin
(41)
This implies that
lim119905rarr119879
minus
1
int
119905
0
119906(120591)2
1198671
119889120591 = +infin (42)
Then the desired assertion immediately follows
Remark 1 In the absence of the viscous term (Δ119906119905) or 119901-
Laplace operator (div(|nabla119906|119901minus2nabla119906)) for the problem (8) theequation reduced to
(1) 119906119905minus Δ119906 + int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(2) 119906119905minus Δ119906 minus Δ119906
119905+ int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(3) 119906119905minusΔ119906+int
119905
0120582(119905minus120591)Δ119906(120591)119889120591 = div(|nabla119906|119901minus2nabla119906)+1199061+120572
and from the process of the proof we can see that the resultsof Theorem 2 still hold
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the NSF of China (40890150 and40890153) and the Scientific Program (2008B080701042) ofGuangdong Province The authors thank the anonymousreferees for their helpful comments and suggestions
References
[1] A B AlrsquoshinMO Korpusov andA G Sveshnikov SveshnikovBlow-up in Nonlinear Sobolev Type Equations De Gruyter 2011
[2] A Y Kolesov E F Mishchenko and N K Rozov ldquoAsymptoticmethods for studying periodic solutions of nonlinear hyper-bolic equationsrdquo Trudy Matematicheskogo Instituta Imeni V ASteklova vol 222 pp 3ndash198 1998
[3] BK Shivamoggi ldquoA symmetric regularized long-wave equationfor shallow water wavesrdquoThe Physics of Fluids vol 29 no 3 pp890ndash891 1986
[4] P Rosenau ldquoEvolution and breaking of the ion-acoustic wavesrdquoPhysics of Fluids vol 31 no 6 pp 1317ndash1319 1988
[5] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119891 (119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[6] M X Wang and X Q Ding ldquoGlobal solutions asymptoticbehavior and blow-up problems for semilinear heat equationsrdquoScience in China A vol 35 pp 1026ndash1034 1992
[7] F Gazzola and T Weth ldquoFinite time blow-up and globalsolutions for semilinear parabolic equations with initial data athigh energy levelrdquo Differential and Integral Equations vol 18no 9 pp 961ndash990 2005
[8] Y C Liu R Z Xu and T Yu ldquoGlobal existence nonexistenceand asymptotic behavior of solutions for the Cauchy problem ofsemilinear heat equationsrdquoNonlinear AnalysisTheory Methodsamp Applications vol 68 no 11 pp 3332ndash3348 2008
[9] H Wang and Y He ldquoOn blow-up of solutions for a semilinearparabolic equation involving variable source and positive initialenergyrdquo Applied Mathematics Letters vol 26 no 10 pp 1008ndash1012 2013
[10] E I Kaikina P I Naumkin and I Shishmarev ldquoA periodicproblem for a Sobolev-type nonlinear equationrdquo FunctionAnalysis and Its Applications vol 44 no 3 pp 14ndash26 2010
[11] Y Cao J X Yin and C H Jin ldquoA periodic problem of asemilinear pseudoparabolic equationrdquo Abstract and AppliedAnalysis vol 2011 Article ID 363579 27 pages 2011
[12] Y Cao J Yin and C Wang ldquoCauchy problems of semilinearpseudo-parabolic equationsrdquo Journal of Differential Equationsvol 246 no 12 pp 4568ndash4590 2009
[13] E I Kaikina ldquoInitial boundary value problems for nonlinearpseudo-parabolic equations in a critical caserdquo Electronic Journalof Differential Equation vol 109 pp 1ndash25 2007
[14] R Xu and J Su ldquoGlobal existence and finite time blow-up fora class of semilinear pseudo-parabolic equationsrdquo Journal ofFunctional Analysis vol 264 no 12 pp 2732ndash2763 2013
[15] M Tsutsumi ldquoExistence and nonexistence of global solutionsfor nonlinear parabolic equationsrdquo Publications of the ResearchInstitute for Mathematical Sciences vol 8 no 2 pp 211ndash2291972
[16] Y C Liu and J S Zhao ldquoNonlinear parabolic equations withcritical initial conditions 119869
(1199060)= 119889 or 119868
(1199060)= 0rdquo Nonlinear
Analysis Theory Methods amp Applications vol 58 no 7-8 pp873ndash883 2004
[17] R Xu X Cao and T Yu ldquoFinite time blow-up and globalsolutions for a class of semilinear parabolic equations at highenergy levelrdquo Nonlinear Analysis Real World Applications vol13 no 1 pp 197ndash202 2012
[18] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[19] G Gripenberg ldquoGlobal existence of solutions of Volterraintegrodifferential equations of parabolic typerdquo Journal of Dif-ferential Equations vol 102 no 2 pp 382ndash390 1993
[20] H Yin ldquoWeak and classical solutions of some nonlinear volterraintegrodifferential equationsrdquo Communications in Partial Dif-ferential Equations vol 17 no 7-8 pp 1369ndash1385 1992
[21] S A Messaoudi ldquoBlow-up of solutions of a semi linear heatequation with a Visco-elastic termrdquo in Nonlinear Elliptic andParabolic Problems H Brezis M Chipot and J Escher Edsvol 64 of Progress in Nonlinear Differential Equations andTheirApplications pp 351ndash356 Birkhauser Basel Switzerland 2005
[22] Y D Shang and B L Guo ldquoOn the problem of the existenceof global solutions for a class of nonlinear convolutionalintegro-differential equations of pseudoparabolic typerdquo ActaMathematicaeApplicatae Sinica vol 26 no 3 pp 512ndash524 2003
[23] Y D Shang and B L Guo ldquoInitial boundary value problemand initial value problem for the nonlinear integro-differentialequations of pseudoparabolic typerdquoMathematicaApplicata vol15 no 1 pp 40ndash45 2002
Abstract and Applied Analysis 7
[24] Y D Shang and B L Guo ldquoInitial-boundary value problemsfor a class of quasilinear integro-differential equations of pseu-doparabolic typerdquo Chinese Journal of Engineering Mathematicsvol 20 no 4 pp 1ndash6 2003
[25] M Ptashnyk ldquoDegenerate quaslinear pseudoparabolic equa-tions with memory terms and variational inequalitiesrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 66 no 12 pp2653ndash2675 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Proof The proof makes use of the so-called ldquoconcavityrdquoarguments For any 119879
0gt 0 let
119872(119905) = int
119905
0
119906(120591)2
2119889120591
+ int
119905
0
nabla119906(120591)2
2119889120591 + (119879
0minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)
(20)
for 119905 isin [0 1198790]
A direct computation yields
1198721015840(119905) = 119906(119905)
2
2+ nabla119906(119905)
2
2
minus (1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)
= 2int
119905
0
(119906 119906119905) 119889120591 + 2int
119905
0
(nabla119906 nabla119906119905) 119889120591
11987210158401015840(119905) = 2 (119906 119906
119905) + 2 (nabla119906 nabla119906
119905)
(21)
By multiplying (8) with 119906 and integrating overΩ
(119906 119906119905) + (nabla119906 nabla119906
119905)
= minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909
(22)
This implies that
11987210158401015840(119905) = minus2nabla119906 (119905)
2
2
minus 2int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus 2nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909
(23)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2
= 2119872 (119905) [minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906(119905)119901
119901+ int
Ω
1199062+120572
119889119909]
minus
120572 + 4
4
[2int
119905
0
(119906 119906119905) 119889120591 + 2int
119905
0
(nabla119906 nabla119906119905) 119889120591]
2
= 2119872 (119905) [minusnabla119906(119905)2
2minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus nabla119906 (119905)119901
119901+ int
Ω
1199062+120572
119889119909] + (120572 + 4)
times 119867 (119905) minus [119872 (119905) minus (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)]
times [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
(24)
where
119867(119905) = [int
119905
0
(119906 119906) 119889120591 + int
119905
0
(nabla119906 nabla119906) 119889120591]
times [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
minus [int
119905
0
(119906 119906119905) 119889120591 + int
119905
0
(nabla119906 nabla119906119905) 119889120591]
2
(25)
Using Schwartzrsquos inequality we have
(int
119905
0(119906 119906119905)119889120591)
2
le int
119905
0
1199062
2119889120591int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
2119889120591
(int
119905
0(nabla119906 nabla119906
119905)119889120591)
2
le int
119905
0
nabla1199062
2119889120591int
119905
0
1003817100381710038171003817nabla119906119905
1003817100381710038171003817
2
2119889120591
int
119905
0
(119906 119906119905) 119889120591int
119905
0
(nabla119906 nabla119906119905) 119889120591
le
1
2
int
119905
0
1199062
2119889120591int
119905
0
1003817100381710038171003817nabla119906119905
1003817100381710038171003817
2
2119889120591
+
1
2
int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
2119889120591int
119905
0
nabla1199062
2119889120591
(26)
By (26) we have
119867(119905) ge 0 119905 isin [0 1198790] (27)
Thus
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120578 (119905) (28)
where
120578 (119905) = minus (120572 + 4) [int
119905
0
(119906119905 119906119905) 119889120591 + int
119905
0
(nabla119906119905 nabla119906119905) 119889120591]
minus 2nabla119906(119905)2
2minus 2int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
minus 2nabla119906(119905)119901
119901+ 2int
Ω
1199062+120572
119889119909
(29)
Abstract and Applied Analysis 5
For the third term on the left of (29) we have
minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (120591) nabla119906 (119905) 119889119909 119889120591
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) nabla [119906 (120591) minus 119906 (119905)] 119889119909 119889120591
+ int
119905
0
120582 (119905 minus 120591) nabla119906(119905)2
2119889120591
(30)
By (29) and (30) we have
120578 (119905) = minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2
+ 2int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) [nabla119906 (120591) minus nabla119906 (119905)] 119889119909 119889120591
minus 2nabla119906119901
119901+ 2int
Ω
1199062+120572
119889119909
ge minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
+ 2int
Ω
1199062+120572
119889119909
minus 2 [
120572 + 2
2
int
119905
0
120582 (119905 minus 120591) nabla119906(120591) minus nabla119906(119905)2
2119889120591
+
1
2 (120572 + 2)
int
119905
0
120582 (119905 minus 120591) nabla119906 (119905)2
2119889120591]
= minus2 (120572 + 2) 119864 (119905) + (120572 + 2) (1 minus int
119905
0
120582 (119905 minus 120591) 119889120591)
times nabla119906 (119905)2
2+
2 (120572 + 2)
119901
nabla119906119901
119901
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
= minus2 (120572 + 2) 119864 (119905) + 120572(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906 (119905)2
2
minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 +
2120572 minus 2119901 + 4
119901
nabla119906119901
119901
(31)
Using Lemma 1 we have
119864 (119905) + int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 lt 119864 (0) (32)
and then
120578 (119905) ge minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572(1 minus int
119905
0
120582 (120591) 119889120591) minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906(119905)2
2
= minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572 minus
120572 (120572 + 2) + 1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906 (119905)2
2
(33)
Since
int
infin
0
120582 (120591) 119889120591 lt
120572 (120572 + 2)
120572 (120572 + 2) + 1
(34)
this implies that
120578 (119905) gt 120575 0 le 119905 lt infin (35)
where 120575 is a positive constantFrom the discussion above we see that
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120575 (36)
From the definition of119872(119905) there exists 120588 gt 0 such that
119872(119905) ge 120588 for 119905 isin [0 119879) (37)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 120588120575 (38)
Thus
(119872(119905)minus1205724
)
10158401015840
= (minus
120572
4
)119872(119905)minus1205724minus2
[11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2]
le (minus
120572
4
) 120588120575119872(119879)minus(120572+8)4
lt 0
(39)
Hence this proves that 119872(119905)minus1205724 reaches 0 in finite time as
119905 rarr 119879minus
1 Since 119879
1is independent of 119879 we may assume that
1198791lt 119879This means
lim119905rarr119879
minus
1
119872(119905) = +infin (40)
6 Abstract and Applied Analysis
or
lim119905rarr119879
minus
1
(int
119905
0
119906(120591)2
2119889120591 + int
119905
0
nabla119906(120591)2
2119889120591
+ (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)) = +infin
(41)
This implies that
lim119905rarr119879
minus
1
int
119905
0
119906(120591)2
1198671
119889120591 = +infin (42)
Then the desired assertion immediately follows
Remark 1 In the absence of the viscous term (Δ119906119905) or 119901-
Laplace operator (div(|nabla119906|119901minus2nabla119906)) for the problem (8) theequation reduced to
(1) 119906119905minus Δ119906 + int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(2) 119906119905minus Δ119906 minus Δ119906
119905+ int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(3) 119906119905minusΔ119906+int
119905
0120582(119905minus120591)Δ119906(120591)119889120591 = div(|nabla119906|119901minus2nabla119906)+1199061+120572
and from the process of the proof we can see that the resultsof Theorem 2 still hold
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the NSF of China (40890150 and40890153) and the Scientific Program (2008B080701042) ofGuangdong Province The authors thank the anonymousreferees for their helpful comments and suggestions
References
[1] A B AlrsquoshinMO Korpusov andA G Sveshnikov SveshnikovBlow-up in Nonlinear Sobolev Type Equations De Gruyter 2011
[2] A Y Kolesov E F Mishchenko and N K Rozov ldquoAsymptoticmethods for studying periodic solutions of nonlinear hyper-bolic equationsrdquo Trudy Matematicheskogo Instituta Imeni V ASteklova vol 222 pp 3ndash198 1998
[3] BK Shivamoggi ldquoA symmetric regularized long-wave equationfor shallow water wavesrdquoThe Physics of Fluids vol 29 no 3 pp890ndash891 1986
[4] P Rosenau ldquoEvolution and breaking of the ion-acoustic wavesrdquoPhysics of Fluids vol 31 no 6 pp 1317ndash1319 1988
[5] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119891 (119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[6] M X Wang and X Q Ding ldquoGlobal solutions asymptoticbehavior and blow-up problems for semilinear heat equationsrdquoScience in China A vol 35 pp 1026ndash1034 1992
[7] F Gazzola and T Weth ldquoFinite time blow-up and globalsolutions for semilinear parabolic equations with initial data athigh energy levelrdquo Differential and Integral Equations vol 18no 9 pp 961ndash990 2005
[8] Y C Liu R Z Xu and T Yu ldquoGlobal existence nonexistenceand asymptotic behavior of solutions for the Cauchy problem ofsemilinear heat equationsrdquoNonlinear AnalysisTheory Methodsamp Applications vol 68 no 11 pp 3332ndash3348 2008
[9] H Wang and Y He ldquoOn blow-up of solutions for a semilinearparabolic equation involving variable source and positive initialenergyrdquo Applied Mathematics Letters vol 26 no 10 pp 1008ndash1012 2013
[10] E I Kaikina P I Naumkin and I Shishmarev ldquoA periodicproblem for a Sobolev-type nonlinear equationrdquo FunctionAnalysis and Its Applications vol 44 no 3 pp 14ndash26 2010
[11] Y Cao J X Yin and C H Jin ldquoA periodic problem of asemilinear pseudoparabolic equationrdquo Abstract and AppliedAnalysis vol 2011 Article ID 363579 27 pages 2011
[12] Y Cao J Yin and C Wang ldquoCauchy problems of semilinearpseudo-parabolic equationsrdquo Journal of Differential Equationsvol 246 no 12 pp 4568ndash4590 2009
[13] E I Kaikina ldquoInitial boundary value problems for nonlinearpseudo-parabolic equations in a critical caserdquo Electronic Journalof Differential Equation vol 109 pp 1ndash25 2007
[14] R Xu and J Su ldquoGlobal existence and finite time blow-up fora class of semilinear pseudo-parabolic equationsrdquo Journal ofFunctional Analysis vol 264 no 12 pp 2732ndash2763 2013
[15] M Tsutsumi ldquoExistence and nonexistence of global solutionsfor nonlinear parabolic equationsrdquo Publications of the ResearchInstitute for Mathematical Sciences vol 8 no 2 pp 211ndash2291972
[16] Y C Liu and J S Zhao ldquoNonlinear parabolic equations withcritical initial conditions 119869
(1199060)= 119889 or 119868
(1199060)= 0rdquo Nonlinear
Analysis Theory Methods amp Applications vol 58 no 7-8 pp873ndash883 2004
[17] R Xu X Cao and T Yu ldquoFinite time blow-up and globalsolutions for a class of semilinear parabolic equations at highenergy levelrdquo Nonlinear Analysis Real World Applications vol13 no 1 pp 197ndash202 2012
[18] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[19] G Gripenberg ldquoGlobal existence of solutions of Volterraintegrodifferential equations of parabolic typerdquo Journal of Dif-ferential Equations vol 102 no 2 pp 382ndash390 1993
[20] H Yin ldquoWeak and classical solutions of some nonlinear volterraintegrodifferential equationsrdquo Communications in Partial Dif-ferential Equations vol 17 no 7-8 pp 1369ndash1385 1992
[21] S A Messaoudi ldquoBlow-up of solutions of a semi linear heatequation with a Visco-elastic termrdquo in Nonlinear Elliptic andParabolic Problems H Brezis M Chipot and J Escher Edsvol 64 of Progress in Nonlinear Differential Equations andTheirApplications pp 351ndash356 Birkhauser Basel Switzerland 2005
[22] Y D Shang and B L Guo ldquoOn the problem of the existenceof global solutions for a class of nonlinear convolutionalintegro-differential equations of pseudoparabolic typerdquo ActaMathematicaeApplicatae Sinica vol 26 no 3 pp 512ndash524 2003
[23] Y D Shang and B L Guo ldquoInitial boundary value problemand initial value problem for the nonlinear integro-differentialequations of pseudoparabolic typerdquoMathematicaApplicata vol15 no 1 pp 40ndash45 2002
Abstract and Applied Analysis 7
[24] Y D Shang and B L Guo ldquoInitial-boundary value problemsfor a class of quasilinear integro-differential equations of pseu-doparabolic typerdquo Chinese Journal of Engineering Mathematicsvol 20 no 4 pp 1ndash6 2003
[25] M Ptashnyk ldquoDegenerate quaslinear pseudoparabolic equa-tions with memory terms and variational inequalitiesrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 66 no 12 pp2653ndash2675 2007
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
For the third term on the left of (29) we have
minus int
Ω
int
119905
0
120582 (119905 minus 120591) Δ119906 (120591) 119889120591119906 (119905) 119889119909
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (120591) nabla119906 (119905) 119889119909 119889120591
= int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) nabla [119906 (120591) minus 119906 (119905)] 119889119909 119889120591
+ int
119905
0
120582 (119905 minus 120591) nabla119906(119905)2
2119889120591
(30)
By (29) and (30) we have
120578 (119905) = minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2
+ 2int
119905
0
120582 (119905 minus 120591) int
Ω
nabla119906 (119905) [nabla119906 (120591) minus nabla119906 (119905)] 119889119909 119889120591
minus 2nabla119906119901
119901+ 2int
Ω
1199062+120572
119889119909
ge minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
+ 2int
Ω
1199062+120572
119889119909
minus 2 [
120572 + 2
2
int
119905
0
120582 (119905 minus 120591) nabla119906(120591) minus nabla119906(119905)2
2119889120591
+
1
2 (120572 + 2)
int
119905
0
120582 (119905 minus 120591) nabla119906 (119905)2
2119889120591]
= minus2 (120572 + 2) 119864 (119905) + (120572 + 2) (1 minus int
119905
0
120582 (119905 minus 120591) 119889120591)
times nabla119906 (119905)2
2+
2 (120572 + 2)
119901
nabla119906119901
119901
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus 2(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906(119905)2
2minus 2nabla119906
119901
119901
= minus2 (120572 + 2) 119864 (119905) + 120572(1 minus int
119905
0
120582 (119905 minus 120591) 119889120591) nabla119906 (119905)2
2
minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591nabla119906 (119905)2
2
minus (120572 + 4) int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 +
2120572 minus 2119901 + 4
119901
nabla119906119901
119901
(31)
Using Lemma 1 we have
119864 (119905) + int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591 lt 119864 (0) (32)
and then
120578 (119905) ge minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572(1 minus int
119905
0
120582 (120591) 119889120591) minus
1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906(119905)2
2
= minus2 (120572 + 2) 119864 (0) + 120572int
119905
0
1003817100381710038171003817119906119905
1003817100381710038171003817
2
1198671119889120591
+ [120572 minus
120572 (120572 + 2) + 1
120572 + 2
int
119905
0
120582 (120591) 119889120591] nabla119906 (119905)2
2
(33)
Since
int
infin
0
120582 (120591) 119889120591 lt
120572 (120572 + 2)
120572 (120572 + 2) + 1
(34)
this implies that
120578 (119905) gt 120575 0 le 119905 lt infin (35)
where 120575 is a positive constantFrom the discussion above we see that
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 119872(119905) 120575 (36)
From the definition of119872(119905) there exists 120588 gt 0 such that
119872(119905) ge 120588 for 119905 isin [0 119879) (37)
and we have
11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2ge 120588120575 (38)
Thus
(119872(119905)minus1205724
)
10158401015840
= (minus
120572
4
)119872(119905)minus1205724minus2
[11987210158401015840(119905)119872 (119905) minus
120572 + 4
4
1198721015840(119905)2]
le (minus
120572
4
) 120588120575119872(119879)minus(120572+8)4
lt 0
(39)
Hence this proves that 119872(119905)minus1205724 reaches 0 in finite time as
119905 rarr 119879minus
1 Since 119879
1is independent of 119879 we may assume that
1198791lt 119879This means
lim119905rarr119879
minus
1
119872(119905) = +infin (40)
6 Abstract and Applied Analysis
or
lim119905rarr119879
minus
1
(int
119905
0
119906(120591)2
2119889120591 + int
119905
0
nabla119906(120591)2
2119889120591
+ (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)) = +infin
(41)
This implies that
lim119905rarr119879
minus
1
int
119905
0
119906(120591)2
1198671
119889120591 = +infin (42)
Then the desired assertion immediately follows
Remark 1 In the absence of the viscous term (Δ119906119905) or 119901-
Laplace operator (div(|nabla119906|119901minus2nabla119906)) for the problem (8) theequation reduced to
(1) 119906119905minus Δ119906 + int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(2) 119906119905minus Δ119906 minus Δ119906
119905+ int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(3) 119906119905minusΔ119906+int
119905
0120582(119905minus120591)Δ119906(120591)119889120591 = div(|nabla119906|119901minus2nabla119906)+1199061+120572
and from the process of the proof we can see that the resultsof Theorem 2 still hold
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the NSF of China (40890150 and40890153) and the Scientific Program (2008B080701042) ofGuangdong Province The authors thank the anonymousreferees for their helpful comments and suggestions
References
[1] A B AlrsquoshinMO Korpusov andA G Sveshnikov SveshnikovBlow-up in Nonlinear Sobolev Type Equations De Gruyter 2011
[2] A Y Kolesov E F Mishchenko and N K Rozov ldquoAsymptoticmethods for studying periodic solutions of nonlinear hyper-bolic equationsrdquo Trudy Matematicheskogo Instituta Imeni V ASteklova vol 222 pp 3ndash198 1998
[3] BK Shivamoggi ldquoA symmetric regularized long-wave equationfor shallow water wavesrdquoThe Physics of Fluids vol 29 no 3 pp890ndash891 1986
[4] P Rosenau ldquoEvolution and breaking of the ion-acoustic wavesrdquoPhysics of Fluids vol 31 no 6 pp 1317ndash1319 1988
[5] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119891 (119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[6] M X Wang and X Q Ding ldquoGlobal solutions asymptoticbehavior and blow-up problems for semilinear heat equationsrdquoScience in China A vol 35 pp 1026ndash1034 1992
[7] F Gazzola and T Weth ldquoFinite time blow-up and globalsolutions for semilinear parabolic equations with initial data athigh energy levelrdquo Differential and Integral Equations vol 18no 9 pp 961ndash990 2005
[8] Y C Liu R Z Xu and T Yu ldquoGlobal existence nonexistenceand asymptotic behavior of solutions for the Cauchy problem ofsemilinear heat equationsrdquoNonlinear AnalysisTheory Methodsamp Applications vol 68 no 11 pp 3332ndash3348 2008
[9] H Wang and Y He ldquoOn blow-up of solutions for a semilinearparabolic equation involving variable source and positive initialenergyrdquo Applied Mathematics Letters vol 26 no 10 pp 1008ndash1012 2013
[10] E I Kaikina P I Naumkin and I Shishmarev ldquoA periodicproblem for a Sobolev-type nonlinear equationrdquo FunctionAnalysis and Its Applications vol 44 no 3 pp 14ndash26 2010
[11] Y Cao J X Yin and C H Jin ldquoA periodic problem of asemilinear pseudoparabolic equationrdquo Abstract and AppliedAnalysis vol 2011 Article ID 363579 27 pages 2011
[12] Y Cao J Yin and C Wang ldquoCauchy problems of semilinearpseudo-parabolic equationsrdquo Journal of Differential Equationsvol 246 no 12 pp 4568ndash4590 2009
[13] E I Kaikina ldquoInitial boundary value problems for nonlinearpseudo-parabolic equations in a critical caserdquo Electronic Journalof Differential Equation vol 109 pp 1ndash25 2007
[14] R Xu and J Su ldquoGlobal existence and finite time blow-up fora class of semilinear pseudo-parabolic equationsrdquo Journal ofFunctional Analysis vol 264 no 12 pp 2732ndash2763 2013
[15] M Tsutsumi ldquoExistence and nonexistence of global solutionsfor nonlinear parabolic equationsrdquo Publications of the ResearchInstitute for Mathematical Sciences vol 8 no 2 pp 211ndash2291972
[16] Y C Liu and J S Zhao ldquoNonlinear parabolic equations withcritical initial conditions 119869
(1199060)= 119889 or 119868
(1199060)= 0rdquo Nonlinear
Analysis Theory Methods amp Applications vol 58 no 7-8 pp873ndash883 2004
[17] R Xu X Cao and T Yu ldquoFinite time blow-up and globalsolutions for a class of semilinear parabolic equations at highenergy levelrdquo Nonlinear Analysis Real World Applications vol13 no 1 pp 197ndash202 2012
[18] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[19] G Gripenberg ldquoGlobal existence of solutions of Volterraintegrodifferential equations of parabolic typerdquo Journal of Dif-ferential Equations vol 102 no 2 pp 382ndash390 1993
[20] H Yin ldquoWeak and classical solutions of some nonlinear volterraintegrodifferential equationsrdquo Communications in Partial Dif-ferential Equations vol 17 no 7-8 pp 1369ndash1385 1992
[21] S A Messaoudi ldquoBlow-up of solutions of a semi linear heatequation with a Visco-elastic termrdquo in Nonlinear Elliptic andParabolic Problems H Brezis M Chipot and J Escher Edsvol 64 of Progress in Nonlinear Differential Equations andTheirApplications pp 351ndash356 Birkhauser Basel Switzerland 2005
[22] Y D Shang and B L Guo ldquoOn the problem of the existenceof global solutions for a class of nonlinear convolutionalintegro-differential equations of pseudoparabolic typerdquo ActaMathematicaeApplicatae Sinica vol 26 no 3 pp 512ndash524 2003
[23] Y D Shang and B L Guo ldquoInitial boundary value problemand initial value problem for the nonlinear integro-differentialequations of pseudoparabolic typerdquoMathematicaApplicata vol15 no 1 pp 40ndash45 2002
Abstract and Applied Analysis 7
[24] Y D Shang and B L Guo ldquoInitial-boundary value problemsfor a class of quasilinear integro-differential equations of pseu-doparabolic typerdquo Chinese Journal of Engineering Mathematicsvol 20 no 4 pp 1ndash6 2003
[25] M Ptashnyk ldquoDegenerate quaslinear pseudoparabolic equa-tions with memory terms and variational inequalitiesrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 66 no 12 pp2653ndash2675 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
or
lim119905rarr119879
minus
1
(int
119905
0
119906(120591)2
2119889120591 + int
119905
0
nabla119906(120591)2
2119889120591
+ (1198790minus 119905) (
1003817100381710038171003817nabla1199060
1003817100381710038171003817
2
2+10038171003817100381710038171199060
1003817100381710038171003817
2
2)) = +infin
(41)
This implies that
lim119905rarr119879
minus
1
int
119905
0
119906(120591)2
1198671
119889120591 = +infin (42)
Then the desired assertion immediately follows
Remark 1 In the absence of the viscous term (Δ119906119905) or 119901-
Laplace operator (div(|nabla119906|119901minus2nabla119906)) for the problem (8) theequation reduced to
(1) 119906119905minus Δ119906 + int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(2) 119906119905minus Δ119906 minus Δ119906
119905+ int
119905
0120582(119905 minus 120591)Δ119906(120591)119889120591 = 119906
1+120572
(3) 119906119905minusΔ119906+int
119905
0120582(119905minus120591)Δ119906(120591)119889120591 = div(|nabla119906|119901minus2nabla119906)+1199061+120572
and from the process of the proof we can see that the resultsof Theorem 2 still hold
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the NSF of China (40890150 and40890153) and the Scientific Program (2008B080701042) ofGuangdong Province The authors thank the anonymousreferees for their helpful comments and suggestions
References
[1] A B AlrsquoshinMO Korpusov andA G Sveshnikov SveshnikovBlow-up in Nonlinear Sobolev Type Equations De Gruyter 2011
[2] A Y Kolesov E F Mishchenko and N K Rozov ldquoAsymptoticmethods for studying periodic solutions of nonlinear hyper-bolic equationsrdquo Trudy Matematicheskogo Instituta Imeni V ASteklova vol 222 pp 3ndash198 1998
[3] BK Shivamoggi ldquoA symmetric regularized long-wave equationfor shallow water wavesrdquoThe Physics of Fluids vol 29 no 3 pp890ndash891 1986
[4] P Rosenau ldquoEvolution and breaking of the ion-acoustic wavesrdquoPhysics of Fluids vol 31 no 6 pp 1317ndash1319 1988
[5] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906
119905=
minus119860119906 + 119891 (119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973
[6] M X Wang and X Q Ding ldquoGlobal solutions asymptoticbehavior and blow-up problems for semilinear heat equationsrdquoScience in China A vol 35 pp 1026ndash1034 1992
[7] F Gazzola and T Weth ldquoFinite time blow-up and globalsolutions for semilinear parabolic equations with initial data athigh energy levelrdquo Differential and Integral Equations vol 18no 9 pp 961ndash990 2005
[8] Y C Liu R Z Xu and T Yu ldquoGlobal existence nonexistenceand asymptotic behavior of solutions for the Cauchy problem ofsemilinear heat equationsrdquoNonlinear AnalysisTheory Methodsamp Applications vol 68 no 11 pp 3332ndash3348 2008
[9] H Wang and Y He ldquoOn blow-up of solutions for a semilinearparabolic equation involving variable source and positive initialenergyrdquo Applied Mathematics Letters vol 26 no 10 pp 1008ndash1012 2013
[10] E I Kaikina P I Naumkin and I Shishmarev ldquoA periodicproblem for a Sobolev-type nonlinear equationrdquo FunctionAnalysis and Its Applications vol 44 no 3 pp 14ndash26 2010
[11] Y Cao J X Yin and C H Jin ldquoA periodic problem of asemilinear pseudoparabolic equationrdquo Abstract and AppliedAnalysis vol 2011 Article ID 363579 27 pages 2011
[12] Y Cao J Yin and C Wang ldquoCauchy problems of semilinearpseudo-parabolic equationsrdquo Journal of Differential Equationsvol 246 no 12 pp 4568ndash4590 2009
[13] E I Kaikina ldquoInitial boundary value problems for nonlinearpseudo-parabolic equations in a critical caserdquo Electronic Journalof Differential Equation vol 109 pp 1ndash25 2007
[14] R Xu and J Su ldquoGlobal existence and finite time blow-up fora class of semilinear pseudo-parabolic equationsrdquo Journal ofFunctional Analysis vol 264 no 12 pp 2732ndash2763 2013
[15] M Tsutsumi ldquoExistence and nonexistence of global solutionsfor nonlinear parabolic equationsrdquo Publications of the ResearchInstitute for Mathematical Sciences vol 8 no 2 pp 211ndash2291972
[16] Y C Liu and J S Zhao ldquoNonlinear parabolic equations withcritical initial conditions 119869
(1199060)= 119889 or 119868
(1199060)= 0rdquo Nonlinear
Analysis Theory Methods amp Applications vol 58 no 7-8 pp873ndash883 2004
[17] R Xu X Cao and T Yu ldquoFinite time blow-up and globalsolutions for a class of semilinear parabolic equations at highenergy levelrdquo Nonlinear Analysis Real World Applications vol13 no 1 pp 197ndash202 2012
[18] S AMessaoudi ldquoA note on blow up of solutions of a quasilinearheat equation with vanishing initial energyrdquo Journal of Mathe-matical Analysis and Applications vol 273 no 1 pp 243ndash2472002
[19] G Gripenberg ldquoGlobal existence of solutions of Volterraintegrodifferential equations of parabolic typerdquo Journal of Dif-ferential Equations vol 102 no 2 pp 382ndash390 1993
[20] H Yin ldquoWeak and classical solutions of some nonlinear volterraintegrodifferential equationsrdquo Communications in Partial Dif-ferential Equations vol 17 no 7-8 pp 1369ndash1385 1992
[21] S A Messaoudi ldquoBlow-up of solutions of a semi linear heatequation with a Visco-elastic termrdquo in Nonlinear Elliptic andParabolic Problems H Brezis M Chipot and J Escher Edsvol 64 of Progress in Nonlinear Differential Equations andTheirApplications pp 351ndash356 Birkhauser Basel Switzerland 2005
[22] Y D Shang and B L Guo ldquoOn the problem of the existenceof global solutions for a class of nonlinear convolutionalintegro-differential equations of pseudoparabolic typerdquo ActaMathematicaeApplicatae Sinica vol 26 no 3 pp 512ndash524 2003
[23] Y D Shang and B L Guo ldquoInitial boundary value problemand initial value problem for the nonlinear integro-differentialequations of pseudoparabolic typerdquoMathematicaApplicata vol15 no 1 pp 40ndash45 2002
Abstract and Applied Analysis 7
[24] Y D Shang and B L Guo ldquoInitial-boundary value problemsfor a class of quasilinear integro-differential equations of pseu-doparabolic typerdquo Chinese Journal of Engineering Mathematicsvol 20 no 4 pp 1ndash6 2003
[25] M Ptashnyk ldquoDegenerate quaslinear pseudoparabolic equa-tions with memory terms and variational inequalitiesrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 66 no 12 pp2653ndash2675 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
[24] Y D Shang and B L Guo ldquoInitial-boundary value problemsfor a class of quasilinear integro-differential equations of pseu-doparabolic typerdquo Chinese Journal of Engineering Mathematicsvol 20 no 4 pp 1ndash6 2003
[25] M Ptashnyk ldquoDegenerate quaslinear pseudoparabolic equa-tions with memory terms and variational inequalitiesrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 66 no 12 pp2653ndash2675 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of