asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents

This article was downloaded by: [University of Arizona]On: 17 June 2013, At: 03:50Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Applicable Analysis: An InternationalJournalPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gapa20

Asymptotic analysis for blow-upsolutions in parabolic equationsinvolving variable exponentsFengjie Li a & Bingchen Liu aa College of Mathematics and Computational Science, ChinaUniversity of Petroleum, Dongying, Shandong Province 257061,P.R. ChinaPublished online: 14 Nov 2011.

To cite this article: Fengjie Li & Bingchen Liu (2013): Asymptotic analysis for blow-up solutions inparabolic equations involving variable exponents, Applicable Analysis: An International Journal,92:4, 651-664

To link to this article: http://dx.doi.org/10.1080/00036811.2011.632767

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.

http://www.tandfonline.com/loi/gapa20

http://dx.doi.org/10.1080/00036811.2011.632767

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Applicable Analysis

Asymptotic analysis for blow-up solutions in parabolic equations

involving variable exponents

Fengjie Li and Bingchen Liu*

College of Mathematics and Computational Science, China University of Petroleum,Dongying, Shandong Province 257061, P.R. China

Communicated by H. Levine

(Received 26 November 2010; final version received 13 October 2011)

In this article, we consider non-negative solutions of the homogeneousDirichlet problems of parabolic equations with local or nonlocalnonlinearities, involving variable exponents. We firstly obtain the necessaryand sufficient conditions on the existence of blow-up solutions, and alsoobtain some Fujita-type conditions in bounded domains. Secondly, theblow-up rates are determined, which are described completely by themaximums of the variable exponents. Thirdly, we show that the blow-upoccurs only at a single point for the equations with local nonlinearities, andin the whole domain for nonlocal nonlinearities.

Keywords: variable exponent; blow-up rate; blow-up set

AMS Subject Classifications: Primary: 35K55, 35B40, 35K15, 35B33

1. Introduction

In this article, we consider the asymptotic blow-up properties of non-negativesolutions for parabolic equations with exponential nonlinearities

utðx, tÞ ¼ Duðx, tÞ þ expfpðxÞuðx, tÞg, ðx, tÞ 2�� ð0,T Þ, ð1:1Þ

or utðx, tÞ ¼ Duðx, tÞ þZ

�

exp�pðxÞuðx, tÞ

�dx, ðx, tÞ 2�� ð0,T Þ, ð1:2Þ

subject to homogeneous Dirichlet boundary conditions, where ��RN is a boundeddomain with a smooth boundary; variable pðxÞ: ��! R is smooth; denotep�¼ inf�p(�), pþ¼ sup�p(�); initial datum u(x, 0)¼ u0(x) is non-negative, non-trivialand smooth, vanishing on the boundary. The existence and uniqueness of classicalnon-negative solutions can be obtained by [1] and the comparison principle is alsotrue. Nonlinear parabolic problems with variable exponents come from severalbranches of applied mathematics and physics, such as flows of electro-rheological or

*Corresponding author. Email: [email protected]

� 201 Taylor & Francis

http://dx.doi.org/10.1080/00036811.2011.632767Vol. 92, No. 4, 651–664,, 2013

3

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

thermo-rheological fluids [2–4], and the processing of digital images [5–7]. For more

detailed information, readers can refer to the books [8,9].Some recent papers [10–12] have studied blow-up questions with variable

exponents, concerning the relationships between the variable exponents and blow-up

versus global existence criteria (i.e. the critical exponents problems). Some problems

are left open, for example, how do blow-up rates depend on the variable exponents?

what is blow-up set made up of? In 2009, Pinasco [10] considered the following

variable parabolic problems

ut ¼ Duþ f ðuÞ, ðx, tÞ 2�� ð0,T Þ,u ¼ 0, ðx, tÞ 2 @�� ð0,T Þ,uðx, 0Þ ¼ u0ðxÞ, x2�,

8<: ð1:3Þ

where ��RN is a bounded domain with a smooth boundary, and the nonlinearities

are the forms

f ðuÞ ¼ aðxÞupðxÞðx, tÞ or f ðuÞ ¼ aðxÞ

Z�

upðxÞðx, tÞdx

with 1< p�� p(x)� pþ<þ1 and 0< c�� a(x)� cþ<þ1. T is the maximal

existence time of solutions. They proved that lim supt!T kuð�, tÞkL1ð�Þ ¼ þ1 if the

initial data are sufficiently large.Antontsev and Shmarev [11] discussed the p(x)-Laplace parabolic equation

ut ¼ divðaðx, tÞjrujpðxÞ�2ruÞ þ bðx, tÞjuj�ðx,tÞ�2u, ðx, tÞ 2�� ð0,T Þ,

subject to null Dirichlet boundary conditions, with variable functions p(x),

�(x, t)2 (1,þ1). If p(x)� 2, a(x, t)� 1, and b(x, t)� b�> 0 (i.e. the semilinear

equation), blow-up happens if the initial data are sufficiently large and

either minx2�

�ðx, tÞ ¼ ��ðtÞ4 2 for all t4 0,

or ��ðtÞ � 2, ��ðtÞ & 2 as t!1 and

Z 11

esð2��ðsÞÞ ds51:

For the evolution p(x)-Laplace equation with the exponents p(x) and �(x), theyproved that every solution, corresponding to sufficiently large initial data, exhibits

blow-up if

bðx, tÞ � b�4 0, atðx, tÞ � 0, btðx, tÞ � 0,

minx2�

�ðxÞ4 2, maxx2�

pðxÞ � minx2�

�ðxÞ:

In [12], Ferreira et al. discussed (1.3) with f (u)¼ up(x) and the corresponding

Cauchy problem in RN. They obtained some interesting results for non-negative p(x)

as follows, for �¼Rn or bounded �, if pþ> 1, there exist blow-up solutions, while if

pþ� 1, then every solution is global. For the Cauchy problem, if p�> 1þ 2/N, there

exist global non-trivial solutions; if 1< p�< pþ� 1þ 2/N, all solutions blow up; if

p�< 1þ 2/N< pþ, there are functions p(x) such that the problem possesses global

non-trivial solutions and functions p(x) such that all solutions blow up. Two more

results of global solutions were obtained: if ��Br(x0) for some x02RN and

r5ffiffiffiffiffiffiffi2Np

, then the problem possesses global non-trivial solutions, regardless of the

2 F. Li and B. Liu652

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

exponent p(x); if p�> 1, then there are global solutions, regardless of the size of �.

The authors in [12] found out some new phenomena in bounded domains, which are

quite different from the corresponding parabolic problems without variable

exponents: There are functions p(x) and bounded domains � such that positive

solutions blow up in finite time for any initial data.By the way, the homogeneous Dirichlet problems for parabolic equations

ut ¼ Duþ vq2ðxÞ, vt ¼ Dvþ uq1ðxÞ, ðx, tÞ 2�� ð0,T Þ,

have been firstly discussed by Bai and Zheng [13]. Some criteria are established for

distinguishing global and non-global solutions of the problem, depending or

independent on initial data. Especially, some Fujita-type result is obtained: there

exist suitable domain � and variable exponents such that any solution blows up in

finite time, just as that in [12].It is worth mentioning that the substitution u¼ log v transforms (1.1) or (1.2) into

the equations with convection terms and power-type nonlinearities:

vt ¼ Dv�jrvj2

vþ vpðxÞþ1, ðx, tÞ 2�� ð0,T Þ, ð1:4Þ

or vt ¼ Dv�jrvj2

vþ v

Z�

vpðxÞ dx, ðx, tÞ 2�� ð0,T Þ, ð1:5Þ

subject to Dirichlet boundary v(x, t)¼ 1 for (x, t)2 @�� (0,T ), with the initial data

v(x, 0)¼ exp{u(x, 0)} for x2�. The classical solution v(�1) exists locally in time and

the local or nonlocal nonlinearities will lead to blow-up of solutions. To the best of

our knowledge, there are no results for the blow-up solutions of (1.4) or (1.5). The

involved convection terms and the nonhomogeneous Dirichlet boundary conditions

make the blow-up criteria in the existing literature [10–12] not be useful to (1.4) or

(1.5) by the comparison principle.This article is arranged as follows. In the following section, the necessary and

sufficient conditions for the existence of blow-up are obtained for (1.1) and (1.2),

respectively, also with some Fujita-type blow-up phenomena in the bounded domain.

In Section 3, we show that the blow-up rates are determined completely by the

maximums of the variables, and the blow-up set is made up of a single point for (1.1)

and the whole domain for (1.2). The parallel results for problem (1.3) are given in

Section 4.

2. Blow-up vs. global existence for (1.1) and (1.2)

The following result shows the existence of blow-up solutions, regardless of the

size of �.

THEOREM 2.1 For problem (1.1), if pþ> 0, blow-up solutions exist for large initial

data, while if pþ� 0, every solution remains global.

We also obtain the Fujita-type blow-up phenomena for (1.1) in bounded

domain �.

Applicable Analysis 3653

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

THEOREM 2.2 There exist suitable positive p(x) and suitable � such that solutions

of (1.1) blow up in finite time for any initial data.

THEOREM 2.3 For problem (1.2), the following results hold:

(i) For any bounded �, if pþ> 0, then there exist blow-up solutions.(ii) For any bounded �, if pþ� 0, every solution remains globally bounded.(iii) If pþ> 0, there exists some suitable ��RN such that globally bounded

solutions exist.

Proof of Theorem 2.1

. If pþ> 0, then blow-up solutions of (1.1) exist for large initial data.

Due to the continuity of p(x), there exists a ball B��, where pþ� p(x)� �> 0.

Introduce a function

�ðtÞ ¼

ZB

’1ðxÞuðx, tÞdx,

where ’1 and �1 are the first eigenfunction and the first eigenvalue of the Dirichlet

Laplacian in B withRB’1(x)dx¼ 1. It is easy to see that �ðtÞ � kuð�, tÞkL1ðBÞ �

kuð�, tÞkL1ð�Þ. We only need to prove that �(t) blows up in finite time. One can

obtain that

�0ðtÞ � ��1�ðtÞ þ

ZB

’1ðxÞ exp��uðx, tÞ

�dx

� ��1�ðtÞ þ�

�þ 1

� ��þ1��þ1ðtÞ,

if �ð0Þ ¼RB ’1ðxÞu0ðxÞdx � maxf1, �

1�

1ð�þ1� Þ

�þ1� g. Hence there exists constant c> 0 such

that �0(t)� c��þ1(t). By integrating this inequality from 0 to t, one can obtain

�ðtÞ � ��ð0Þ ��

ct

� ��1=�: ð2:1Þ

So �(t) blows up in finite time for positive �. Hence u blows up in finite time for

pþ> 0 and largeRB’1(x)u0(x)dx.

. If pþ� 0, then every solution of (1.1) remains global.

It is easy to check that every positive solution of the linear problem remains

global:

�ut ¼ D �uþ 1, ðx, tÞ 2�� ð0,T Þ,

�u ¼ 0, ðx, tÞ 2 @�� ð0,T Þ,

�uðx, 0Þ ¼ u0ðxÞ, x2�,

8><>:

where u0(x) is the initial datum of (1.1).For any p(x)� pþ� 0,

Duþ expfpðxÞug � Duþ 1, ðx, tÞ 2�� ð0,T Þ:

By the comparison principle, �u is a global super solution for (1.1).Theorem 2.1 has been proved. g


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

Remark 2.1 By (2.1), one can check that larger u0 (largerRB ’1ðxÞu0ðxÞdx) leads to

smaller blow-up time of solutions for (1.1). g

Secondly, we prove some Fujita-type blow-up phenomena in bounded �.

Proof of Theorem 2.2 Consider the following parabolic problem

ut ¼ Duþ u �pðxÞ, ðx, tÞ 2�� ð0,T Þ,u ¼ 0, ðx, tÞ 2 @�� ð0,T Þ,uðx, 0Þ ¼ u0ðxÞ, x2�,

8<: ð2:2Þ

where function �pðxÞ is positive, smooth and bounded in �� and u0 is defined in (1.1).By Theorem 3.7 in [12], for an arbitrary function q(x), defined in the unit ball

B1(0) verifying that q(x)� 1 changes sign, there exists L> 0 sufficiently large such

that, if the ball BL(x0)��, then the solution to (2.2) blows up in finite time for any

non-trivial non-negative initial data u0, being �pðxÞ � qððx� x0Þ=LÞ, for every

x2BL(x0).One can obtain by the comparison principle that u� 0 for any non-negative

non-trivial initial data u0. Moreover, for the positive �pðxÞ defined above, there exists

ut ¼ Duþ exp�

�pðxÞu�� Duþ u �pðxÞ, ðx, tÞ 2�� ð0,T Þ:

By the comparison principle, every positive solution of (2.2) is the blow-up

subsolution of (1.1) for any non-negative non-trivial initial data. Hence, every

positive solution of (1.1) blows up in finite time. g

Proof of Theorem 2.3 (i) It can be proved by the similar methods to establish

Theorem 2.1. We omit the details here. g

(ii) It is easy to check that every solution of the following linear parabolic problem

remains globally bounded:

�ut ¼ D �uþ j�j, ðx, tÞ 2�� ð0,T Þ,�u ¼ 0, ðx, tÞ 2 @�� ð0,T Þ,�uðx, 0Þ ¼ u0ðxÞ, x2�,

8<: ð2:3Þ

where u0(x) is the initial datum of (1.2). In fact, one can check that, if the constant

A> 0 is large enough, the function A (x) can be a globally bounded super solution

of problem (2.3), where (x) satisfies that

�D ¼ 1, x2�, ¼ 0, x2 @�:

�ð2:4Þ

For any p(x)� pþ� 0, we obtain that

ut � DuþZ

�

exp�pþuðx, tÞ

�dx � Duþ j�j, ðx, tÞ 2�� ð0,T Þ:

By the comparison principle, �u is a global bounded super solution of u. g

(iii) For pþ> 0, if j�j< exp{�pþ} and the diameter of � is less than 2ffiffiffiffiffiffiffi2Np

, then

there exist some x02RN and positive constant R5

ffiffiffiffiffiffiffi2Np

such that ��BR(x0).


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

Introduce a function

wðxÞ ¼2N� jx� x0j

2

2N, x2BRðx0Þ:

It is easy to check that 0<w(x)� 1 for x2 �� and

�DwðxÞ ¼ 14 expfpþgj�j �

Z�

exp�pðxÞwðxÞ

�dx, x2�:

By the comparison principle, w(x) can be a bounded super solution of (1.3) for pþ> 0and small initial data. g

3. Blow-up rate and blow-up set for (1.1) and (1.2)

In this section, we deal with asymptotic properties of blow-up solutions. Let ’ and �denote the first eigenfunction and the first eigenvalue of the Dirichlet Laplacian in �,normalized by

R�’(x)dx¼ 1, respectively.

THEOREM 3.1 Let p(x)> 0, x2�.

(i) (Lower blow-up rate estimate) Assume

ðH Þ Du0ðxÞ þ exp�pðxÞu0ðxÞ

�� 0, x2�:

Then there exists some constant c> 0 such that the blow-up solutions of (1.1)satisfy that

exp�kuð�, tÞkL1ð�Þ

�� cðT� tÞ�1=pþ :

(ii) (Upper blow-up rate estimate) Assume u0, p radial with u00, p0 � 0, Dp� 0 in

�¼BR¼ {x2RNjjxj ¼ r<R} and

Du0ðxÞ þ ð1� "’ðxÞÞ exp�pðxÞu0ðxÞ

�� 0, x2BR:

Then, for some constant C> 0 and small "> 0,


�� CðT� tÞ�1=pþ :

(iii) (Single point blow-up) Assume u0, p radial with u00, p0 � 0, Dp� 0 in �¼BR

and let (H) be satisfied. Then the blow-up occurs at a single point x¼ 0.

THEOREM 3.2 For (1.2) with p(x)> 0 in �, the solutions blow up everywhere in thewhole domain �. Moreover, there exists constant c> 0 such that

expfuðx, tÞg � cðT� tÞ�1=pþ , ðx, tÞ 2K� � ½T=2,T Þ

with K� ¼ fy2�: distð y, @�Þ � �4 0g. If the measure of the sub-domain of � suchthat p(x)� pþ is not zero, then

expfuðx, tÞg � CðT� tÞ�1=pþ , ðx, tÞ 2�� ½T=2,T Þ:

The blow-up rates for parabolic problems (1.1) and (1.2) with variable exponentsdepend sensitively on the maximums pþ of the variable exponents p(x), which can be


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

understood that the asymptotic blow-up properties of solutions rely on the maximal

capabilities of the nonlinearities. It can be found that the presentations of blow-up

rates are just the ones of parabolic systems without variable exponents [14–16]

provided p(x)� pþ. Moreover, the radial solutions blow up at a single point, also

compatible with the parabolic systems without variable exponents (see, e.g. [14,15]).

To the best of our knowledge, Theorems 3.1 and 3.2 have been the first results about

asymptotic blow-up properties for parabolic problems involving variable exponents.

Proof of Theorem 3.1 We firstly prove the lower bound of blow-up solutions.

Since the initial data satisfy that

Du0ðxÞ þ exp�pðxÞu0ðxÞ

�� 0, x2 ��,

from which we deduce that ut(x, t)� 0 for (x, t)2�� [0,T ) by the comparison

principle.By Green’s identity, we have

uðx, tÞ ¼

Z�

�ðx, y, t, zÞuð y, zÞdyþ

Z t

z

Z�

�ðx, y, t, �Þ exp�pð yÞuð y, �Þ

�dy d�

þ

Z t

z

Z@�

�ðx, y, t, �Þ@u

@�dSy d�

� kuð�, zÞkL1ð�Þ þ ðt� zÞ exp�pþkuð�, tÞkL1ð�Þ

�,

where � is the fundamental solution of the heat equations [1,8], i.e.

�ðx, y, t, �Þ ¼1

½4�ðt� �ÞN=2exp �

jx� yj2

4ðt� �Þ

� �:

Hence,

kuð�, tÞkL1ð�Þ � kuð�, zÞkL1ð�Þ þ CðT� zÞ exp�pþkuð�, tÞkL1ð�Þ

�: ð3:1Þ

For the blow-up of u and any z2 (0,T ), one can choose t2 (z,T ) such that

ku(�, t)k1¼ku(�, z)k1þ 1. By (3.1), u satisfies that

1 � CðT� zÞ exp�pþkuð�, zÞkL1ð�Þ þ pþ

�:

Then there exists some positive constant c such that


�� cðT� tÞ�1=pþ , t2 ð0,T Þ:

Secondly, we show the upper bound of blow-up solutions. Define a function

Jðx, tÞ ¼ utðx, tÞ � "ðx, tÞ expfpðxÞuðx, tÞg, ðx, tÞ 2�� ½0,T Þ,

where (x, t)¼ exp{��t}’(x). Considering the assumptions on u0 and p, one can

obtain that

rp � ru � 0, DpðxÞ � 0, pð0Þ ¼ pþ, ut � 0, ðx, tÞ 2�� ð0,T Þ:


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

It can be checked that

Jtðx, tÞ � DJðx, tÞ � pðxÞ expfpðxÞuðx, tÞgJðx, tÞ, ðx, tÞ 2�� ð0,T Þ,

Jðx, 0Þ � 0, x2�,

Jðx, tÞ � 0, ðx, tÞ 2 @�� ð0,T Þ:

By the comparison principle, we have

utðx, tÞ � "ðx, tÞ expfpðxÞuðx, tÞg, ðx, tÞ 2�� ½0,T Þ:

Hence, ut(0, t)� "(0,T )exp{p(0)u(0, t)}, t2 [0,T ). The upper bound follows.At last, we determine the blow-up set of radially decreasing solutions. We discuss

it by contradiction. If the solutions did not blow up only at a single point, then there

would exist some r02 (0,R) such that

lim supt!T

uðr0, tÞ ¼ þ1:

Hence, u(r, t) can blow up at time T for r2 [0, a]� [0, r0). Introduce a function

Hðr, tÞ ¼ urðr, tÞ þ "cðrÞumðr, tÞ, ðr, tÞ 2 ð0, aÞ � ð0,T Þ,

where m> 1 and c(r)¼ sin2(�r/a). By a simple computation, there is

Ht�Hrr�N�1

rHr�

pðrÞexpfpðrÞug�2"c0ðrÞmum�1

H

þ "cðrÞum

"expfpðrÞug

mu�pðrÞ

þ2"c0ðrÞmum�1

�N�1

r

c0ðrÞ

cðrÞ�c00ðrÞ

cðrÞ

#, ðr,tÞ2ð0,aÞ� ð0,T Þ:

Considering the requirements on u0, we obtain that there exist some � 2 (0,T ) and

small "> 0 such that

exp�pðrÞuðr, tÞ

� m

uðr, tÞ� pðrÞ

þ 2"c0ðrÞmum�1ðr, tÞ �

N� 1

r

c0ðrÞ

cðrÞ�c00ðrÞ

cðrÞ� 0,

ðr, tÞ 2 ð0, aÞ � ð�,T Þurðr, �Þ

þ "cðrÞumðr, �Þ � 0, r2 ð0, aÞ:

In fact, this follows from ur< 0 in (0,R)� (0,T ) and urr< 0 in (0,T ), which are

consequences of the strong maximum principle and of the Hopf lemma. Hence,

we have

Ht �Hrr �N� 1

rHr �

pðrÞ expfpðrÞug � 2"c0ðrÞmum�1

H, ðr, tÞ 2 ð0, aÞ � ð�,T Þ,

Hðr, tÞ ¼ 0, t2 ð�,T Þ, r ¼ 0, a,

Hðr, �Þ � 0, r2 ð0, aÞ:

By the comparison principle, there is

urðr, tÞ � �"cðrÞumðr, tÞ, ðr, tÞ 2 ð0, aÞ � ð�,T Þ:


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

By integration, one can obtain that

u1�mðr, tÞ � c

Z r

0

cðsÞds, ðr, tÞ 2 ð0, aÞ � ð�,T Þ, m4 1,

a contradiction with the blow-up of u in (0, a). g

In order to prove Theorem 3.2, we introduce two lemmas at first. Forconvenience, define

GðtÞ ¼

Z t

0

Z�


�dx dt and HðtÞ ¼

Z t

0

GðsÞds:

LEMMA 3.1 Let u be a blow-up solution of (1.2). Then the following results hold withconstant C> 0,

(a) u(x, t)�CþG(t) for (x, t)2�� [T/2, T );

(b) limt!Tuðx, tÞGðtÞ ¼ limt!T

kuð�, tÞkL1ð�ÞGðtÞ ¼ 1, uniformly on any compact subset of �;

(c) uðx, tÞ � GðtÞ � C�Nþ1ð1þHðtÞÞ for (x, t)2K�� [T/2,T ) with K� defined in

Theorem 3.2.

Proof Cases (a), (b) and (c) of Lemma 3.1 can be obtained by Lemma 4.4,Theorem 4.1 and Lemma 4.5 in [17], respectively. We omit the details here. g

LEMMA 3.2 Under the conditions of Theorem 3.2, there exist positive constants c andC such that

c exp�pþGðtÞ

�� G0ðtÞ � C exp

�pþGðtÞ

�:

Proof By Lemma 3.1(a), one can check that


�� exp

�pþðCþ GðtÞÞ

�� C exp

�pþGðtÞ

�ð3:2Þ

for (x, t)2�� [T/2, T ). Integrating (3.2) over �, we have

G0ðtÞ �

Z�

exp�pþðCþ GðtÞÞ

�dx � C exp

�pþGðtÞ

�, t2 ½T=2,T Þ:

On the other hand, by Lemma 3.1 (c), there is

G0ðtÞ �

ZK�

exp p� GðtÞ �C

�Nþ1ð1þHðtÞÞ

� �� dx

¼ jK�j exp p� GðtÞ �C

�Nþ1ð1þHðtÞÞ

� �� , t2 ½T=2,T Þ: ð3:3Þ

Since G(t)!þ1 monotonically as t!T, it follows with any "> 0 that

HðtÞ

GðtÞ¼

R T�"0 GðsÞdsþ

R tT�" GðsÞds

GðtÞ�

R T�"0 GðsÞds

GðtÞþ ":

Therefore,

limt!T

HðtÞ

GðtÞ¼ 0: ð3:4Þ


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

Considering (3.4), there exists some t1 near T such that

p�GðtÞ �p�C

�Nþ1ð1þHðtÞÞ �

p�2GðtÞ:

Hence,

G0ðtÞ ¼ f ðuÞ � jK�j expp�2GðtÞ

n o, t2 ½t1,T Þ: ð3:5Þ

Integrating (3.5) over (t, s)� (t1, T ), and then letting s!T, we have

2

p�exp �

p�2GðtÞ

n o� jK�jðT� tÞ,

which implies that

GðtÞ �2

p�

nj logðT� tÞj þ log 2� logð p�jK�jÞ

o, t2 ½t1,T Þ:

Hence, HðtÞ ¼R t0 GðsÞds � C0, t2 [0,T ). By (3.3) and the requirement on p(x), for �

small enough, one can obtain that

G0ðtÞ �

ZK�

exp pðxÞ GðtÞ �C

�Nþ1ð1þ C0Þ

� �� dx

¼ C

ZK�

exp pðxÞGðtÞ� �

dx

� c expfpþGðtÞg, t2 ½T=2,T Þ:

This lemma is proved. g

Proof of Theorem 3.2 By Lemma 3.1(b), one can check that the blow-up set is made

up of the whole domain.By Lemma 3.2, one can obtain that there are positive constants c and C such that

cðT� tÞ�1=pþ � expfGðtÞg � CðT� tÞ�1=pþ :

By using (a) of Lemma 3.1,

expfuðx, tÞ � Cg � expfGðtÞg � CðT� tÞ�1=pþ , ðx, tÞ 2�� ½T=2,T Þ:

Then the upper bound of blow-up rate follows. Similarly, by using (c) of Lemma 3.1

along with H(t)�C0 for t2 [0,T ), the lower bound of blow-up rate is obtained. g

4. The parallel blow-up results for (1.3)

The parallel results for Equation (1.3) with f ðuÞ ¼R

� upðxÞðx, tÞdx are obtained.

THEOREM 4.1 For (1.3) with f ðuÞ ¼R

� upðxÞðx, tÞdx, the following results hold:

(i) For any bounded �, if pþ> 1, then there exist blow-up solutions.(ii) For any bounded �, if pþ� 1, every solution remains global.(iii) For any bounded �, if p�> 1, there exist globally bounded solutions.


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

(iv) If pþ> 1, there exists suitable ��RN such that globally bounded solutionsexist.

Remark 4.1 By Theorem 4.1, one can find that, for (1.3) with f ðuÞ ¼R

� upðxÞðx, tÞdx,

. 0� p�� pþ� 1: Regardless of domain � and initial data, every non-negativesolution remains global.

. 1< p�� pþ: Regardless of domain �, both blow-up solutions and globalsolutions coexist.

. 0� p�� 1< pþ: Regardless of domain �, blow-up solutions always exist.For suitable �, globally bounded solutions exist. It is unknown whetherglobal solutions always exist for any �.

g

By the methods to establish Theorems 3.1 and 3.2, we obtain the following tworesults for (1.3).

THEOREM 4.2 Let f (u)¼ up(x)(x, t) and pþ> 1 in (1.3).

(i) (Lower blow-up rate estimate) Assume

ðH0Þ Du0ðxÞ þ upðxÞ0 ðxÞ � 0, x2�:

Then there exists some constant c> 0 such that the blow-up solutions of (1.3)satisfy that

kuð�, tÞkL1ð�Þ � cðT� tÞ� 1

pþ�1:

(ii) (Upper blow-up rate estimate) Assume u0, p radial with u00, p0 � 0, Dp� 0 in

�¼BR¼ {x2RNjjxj ¼ r<R} and

Du0ðxÞ þ ð1� "’ðxÞÞupðxÞ0 ðxÞ � 0, x2BR:

Then, for some constant C> 0 and small "> 0,

kuð�, tÞkL1ð�Þ � CðT� tÞ� 1

pþ�1:

(iii) (Single point blow-up) Assume u0, p> 1 radial with u00, p0 � 0, Dp� 0 in

�¼BR and let (H0) be satisfied. Then the blow-up occurs at a singlepoint x¼ 0.

THEOREM 4.3 For (1.3) with f ðuÞ ¼R

� upðxÞðx, tÞdx and pþ> 1, the solutions blow upeverywhere in the whole domain �.

Moreover, there exists a constant c> 0 such that

uðx, tÞ � cðT� tÞ� 1

pþ�1, ðx, tÞ 2K� � ½T=2,T Þ

with K� ¼ fy2�: distð y, @�Þ � �4 0g. If the measure of the sub-domain of � suchthat p(x)� pþ is not zero, then

uðx, tÞ � CðT� tÞ� 1

pþ�1, ðx, tÞ 2�� ½T=2,T Þ:

Proof for Theorem 4.1 (i) It can be obtained by the similar methods used inTheorem 2.3 (i) of this article. g


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

(ii) Let pþ< 1. Introduce a function

wðtÞ ¼ ku0ð�ÞkL1ð�Þ expftþ tg, t � 0,

where t> 0 satisfies wð0Þ � maxf1, j�j1=ð1�pþÞg. Then one can verify that

w0ðtÞ � ku0ð�ÞkpþL1ð�Þ expfpþðtþ tÞgj�j �

Z�

wpðxÞðtÞdx, x2�, t4 0,

wðtÞ � ku0ð�ÞkL1ð�Þ expftg � u0ðxÞ, x2�,

wðtÞ � 0, x2 @�, t4 0:

By the comparison principle, u remains global for any non-negative non-trivial initialdata. On the other hand, if pþ¼ 1, we introduce another function

�uðx, tÞ ¼ A exp�ltþ ðxÞ

�, ðx, tÞ 2 �� ½0,T Þ,

where A and l are positive constants to be determined and solves problem (2.4).For any bounded domain �, jr j and k ð�ÞkL1ð�Þ remain bounded in ��, and

�u> 0 on @�� [0,T ). One can choose large A and large l, satisfying that

A � max�1, ku0ð�ÞkL1ð�Þ

�,

l4 jr j2 þ j�j exp�k ð�ÞkL1ð�Þ

�� 1

such that �u(x, 0)� u0(x)� 0 for x2 ��, and

�utðx, tÞ � D �uðx, tÞ þ Aj�j exp�ltþ k ð�ÞkL1ð�Þ

�� D �uðx, tÞ þ

Z�

ApðxÞ exp�pðxÞltþ pðxÞ ðxÞ

�dx

¼ D �uðx, tÞ þ

Z�

�upðxÞðx, tÞdx:

By the comparison principle, one can find that, if pþ¼ 1, �u is a global super solutionfor (1.3) with f ðuÞ ¼

R�upðxÞðx, tÞdx. g

(iii) For any bounded �, there exist some x0 and R such that ��BR(x0)�RN.Introduce a function

wðxÞ ¼kðA� jx� x0j

2Þ

A, x2BRðx0Þ

with constant k2 (0, 1) to be determined and constant A>max{2N, R2}. Obviously,w(x)> 0 on @� and 0<w(x)� k< 1 in �.

For any p�> 1, we choose k, satisfying 05 k5 minf1, ð 2NAj�jÞ1=ð p��1Þg, such that

�DwðxÞ �Z

�

wpðxÞðxÞdx, x2�:

By the comparison principle, w(x) is a global super solution of (1.3) withf ðuÞ ¼

R� upðxÞðx, tÞdx under small initial data u0 for p�> 1. g

Remark 4.2 By the similar method used in Theorem 4.1(iii), one can obtain that, ifp�> 1, there exist globally bounded solutions in any � for (1.3) with f (u)¼ up(x)

in [12]. g


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

Proof of Theorem 4.1 (iv) If the diameter of � is less than 2ffiffiffiffiffiffiffi2Np

and j�j5 2�pþ ,then there exist some x02R

N, r �ffiffiffiffiffiffiffi2Np

and a ball Br(x0)�RN such that ��Br(x0).Introduce a function

wðxÞ ¼2N� jx� x0j

2

2Nþ 1, x2Brðx0Þ:

Obviously, w(x)� 1 on @� and 1�w(x)� 2 in �. For pþ> 1, we have

�DwðxÞ ¼ 14 2pþj�j ¼

Z�

2pþ dx �

Z�

wpðxÞðxÞdx, x2�:

By the comparison principle, w(x) is a global super solution of (1.3) withf ðuÞ ¼

R� upðxÞðx, tÞdx under small initial data u0 for pþ> 1. g

Acknowledgements

This article is partially supported by Shandong Provincial Natural Science Foundation, China(Nos. ZR2009AQ016, ZR2010AQ011), and by the Fundamental Research Funds for theCentral Universities. The authors thank the editor and the reviewers for their constructivesuggestions to improve the quality of this article.

References

[1] A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall,

Englewood Cliffs, NJ, 1969.

[2] E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids,

Arch. Ration. Mech. Anal. 164 (2002), pp. 213–259.[3] S.N. Antontsev and J.F. Rodrigues, On stationary thermo-rheological viscous flows,

Ann. Univ. Ferrara, Sez. VII Sci. Mat. 52 (2006), pp. 19–36.[4] K. Rajagopal and M. Ruzicka, Mathematical modelling of electro-rheological fluids,

Contin. Mech. Thermodyn. 13 (2001), pp. 59–78.[5] R. Aboulaicha, D. Meskinea, and A. Souissia, New diffusion models in image processing,

Comput. Math. Appl. 56 (2008), pp. 874–882.

[6] Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image

restoration, SIAM J. Appl. Math. 66 (2006), pp. 1383–1406.[7] S. Levine, Y. Chen, and J. Stanich, Image restoration via nonstandard diffusion,

Department of Mathematics and Computer Science, Duquesne University, 2004, Tech.

Rep. ] 04–01.[8] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.[9] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, and A.P. Mikhailov, Blow-up in

Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, New York, 1995.

[10] J.P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents,

Nonlinear Anal. 71 (2009), pp. 1094–1099.[11] S.N. Antontsev and S. Shmarev, Blow-up of solutions to parabolic equations with

nonstandard growth conditions, J. Comput. Appl. Math. 234 (2010), pp. 2633–2645.[12] R. Ferreira, A. de Pablo, M. Perez-Llanos, and J.D. Rossi, Critical exponents for a

semilinear parabolic equation with variable reaction, to appear in the Royal Society of

Edinburgh Proceedings A (Mathematics). Available at http://mate.dm.uba.ar/�jrossi/

FPPR-px-zamp.pdf.


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

[13] X.L. Bai and S.N. Zheng, A semilinear parabolic system with coupling variable exponents,Ann. Mat. Pura Appl. 190 (2011), pp. 525–537.

[14] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,Indiana Univ. Math. J. 34 (1985), pp. 425–447.

[15] A. Friedman and Y. Giga, A single point blow-up for solutions of semilinear parabolicsystems, J. Fac. Sci. Univ. Tokyo Sect IA Math. 34 (1987), pp. 65–79.

[16] W.X. Liu, The blow-up rate of solutions of semilinear heat equations, J. Diff. Eqns 77

(1989), pp. 104–122.[17] Ph. Souplet, Uniform blow-up profile and boundary behaviour for diffusion equation with

nonlocal nonlinear source, J. Differ. Eqns 153 (1999), pp. 374–406.


Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

3:50

17

June

201

3

asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents

Documents