dissipative parabolic equations in locally uniform spaces

MATHEMATISCHE

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CH

RICH

TEN

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Dissipative parabolic equations in locally uniform spaces

J. M. Arrieta1, J. W. Cholewa2, Tomasz Dlotko2, and A. Rodrıguez–Bernal1

1 Departamento de Matematica Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain2 Institute of Mathematics, Silesian University, 40-007 Katowice, Poland

Received 9 February 2005, accepted 5 January 2006Published online 8 October 2007

Key words Reaction-diffusion equations, Cauchy problem in RN , dissipativeness, global attractor

MSC (2000) 35K15, 35K57, 35B40, 35B41

The Cauchy problem for a semilinear second order parabolic equation ut = ∆u + f(x, u,∇u), (t, x)∈R+ ×

RN , is considered within the semigroup approach in locally uniform spaces W s,p

U

`R

N´. Global solvability,

dissipativeness and the existence of an attractor are established under the same assumptions as for problemsin bounded domains. In particular, the condition sf(s,0) < 0, |s| > s0 > 0, together with gradient’s“subquadratic” growth restriction, are shown to guarantee the existence of an attractor for the above mentionedequation. This result cannot be located in the previous references devoted to reaction-diffusion equations in thewhole of R

N .

Math. Nachr. 280, No. 15, 1643 – 1663 (2007) / DOI 10.1002/mana.200510569

Math. Nachr. 280, No. 15, 1643 – 1663 (2007) / DOI 10.1002/mana.200510569

Dissipative parabolic equations in locally uniform spaces

J. M. Arrieta∗1, J. W. Cholewa∗∗2, Tomasz Dlotko∗∗∗2, and A. Rodrıguez–Bernal†1

1 Departamento de Matematica Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain2 Institute of Mathematics, Silesian University, 40-007 Katowice, Poland

Received 9 February 2005, accepted 5 January 2006Published online 8 October 2007

Key words Reaction-diffusion equations, Cauchy problem in RN , dissipativeness, global attractor

MSC (2000) 35K15, 35K57, 35B40, 35B41

The Cauchy problem for a semilinear second order parabolic equation ut = ∆u + f(x, u,∇u), (t, x)∈R+ ×

RN , is considered within the semigroup approach in locally uniform spaces W s,p

U

`R

N´. Global solvability,

dissipativeness and the existence of an attractor are established under the same assumptions as for problemsin bounded domains. In particular, the condition sf(s,0) < 0, |s| > s0 > 0, together with gradient’s“subquadratic” growth restriction, are shown to guarantee the existence of an attractor for the above mentionedequation. This result cannot be located in the previous references devoted to reaction-diffusion equations in thewhole of R

N .

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction and main results

In this paper we study the Cauchy problem for an autonomous second order semilinear parabolic equation of theform {

ut = ∆u+ f(x, u,∇u), t > 0, x ∈ RN ,

u(0, x) = u0(x), x ∈ RN .

(1.1)

Our goal is to give structure conditions on the nonlinearity guaranteeing global well posedness of (1.1), dissipa-tiveness and the existence of an attractor.

Note that several different versions of (1.1), including the case of systems, have been considered by a numberof authors, e.g. [1, 6, 9, 12, 13, 18, 19, 23, 27, 32, 33, 34], where attractors have been constructed under suitablefunctional settings and suitable growth sign and structure conditions on the nonlinear term f(x, u,∇u).

Here, we give a functional setting in very large spaces and prove the existence of a suitable notion of attractorthat attracts in suitable strong norms.

To illustrate the sort of difficulties that one faces when studying (1.1), consider the bi-stable reaction termf = f(u) = u − u3. Notice that in dimension one this equation has traveling waves of all velocities connectingthe constant equilibria 0 and ±1; [26, page 136]. Also a continuum of bounded space periodic equilibria is easilyshown to exists. Moreover, for any dimension, one can show that for any reasonable initial data the sup normof the solution will be asymptotically bounded by 1. On the other hand for any nonnegative smooth compactlysupported initial data, the solution converges to 1 inL∞

loc

(R

N); see e.g. [6]. Hence to account for all this dynamics

one must choose a large space, containing bounded functions, but whose norm is not too strong. For example,traveling waves do not connect 0 and ±1 in the sup norm.

In the case f depends on the gradient, some subquadratic growth restriction is needed, otherwise solutionsmay blow-up in a finite time in C1 norm, see [41].

∗ e-mail: [email protected] Phone: +34 91 3944232, Fax: +34 91 3944102∗∗ Corresponding author: e-mail: [email protected], Phone: +48 32 2582 976, Fax: +48 32 2582 976∗∗∗ e-mail: [email protected], Phone: +48 32 2582976, Fax: +48 32 2582976† e-mail: [email protected], Phone: +34 91 3944409, Fax: +34 91 3944613

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1644 Arrieta, Cholewa, Dlotko, and Rodrıguez–Bernal: Parabolic equations in locally uniform spaces

Hence, in the cited references different approaches for the functional setting and for the structure conditionsin the nonlinearity have been proposed. For example, weighted Sobolev spaces W k,p

ρ

(R

N)

have been used in[1, 9, 18, 21, 22]. The space of bounded and uniformly continuous functionsBUC

(R

N), was used in [32]. More

recently Bessel potentials spacesHsp

(R

N)

have been used in [6]. The so-called locally uniform spaces have beenused in [12, 13, 33, 34] and a systematic study of linear equations in such spaces was developed in [7]. Hencethis paper is a natural “nonlinear” continuation of that one.

In each one of these settings, the authors impose some structure conditions on the nonlinear terms. For examplein [9, 14] the assumption on the nonlinear term reads

f(x, u) = −λ0u+ f0(u) + g(x), with λ0 > 0, f0(u)u ≤ 0, (1.2)

with g a suitable given function, while in [18] reads (for vector valued solutions)

f(x, u,∇u) = −λ0u+ f0(u,∇u) + g(x), with λ0 > 0, f0(u,∇u)u ≤ 0. (1.3)

Note that these sort of conditions imply strong restrictions on the sign of both the linear and the nonlinear partsof the equation. Slightly weaker assumptions on the linear part are used in [32] and even weaker ones have beenconsidered in [6]. On the other hand assuming monotonicity conditions on f0, conditions on the linear term aredropped in [13, 22], which implies that the bistable equation can be handled within these frameworks. As the twolatter references work in large weighted spaces, weak attractivity properties are obtained. The same happens in[9]. Also, note that [9, 21, 22] do not allow the term f0 to depend on x.

As mentioned before, our goal here is to give a functional setting in a very large space and prove the existenceof a suitable notion of attractor that attracts in appropriately strong norms.

In particular, our results show that if we consider the particular case of (1.1)

ut = ∆u+ f(u,∇u), t > 0, x ∈ RN , u(0, x) = u0(x), x ∈ R

N , (1.4)

where the nonlinear term does not depend on x and grows subquadratically with respect to ∇u, then this problemhas an attractor iff the associated ordinary differential equation for spatially homogeneous solutions

U = f(U, 0), t > 0, U(0) = U0 ∈ R, (1.5)

has an attractor. This, in turn, reduces to the condition

sf(s, 0) < 0, |s| > s0 > 0.

In this paper we will use locally uniform spaces as phase spaces for (1.1). These are the Banach spacesconsisting of all (equivalence classes of ) measurable functions φ : R

N → R satisfying

‖φ‖LpU (RN ) := sup

y∈RN

‖φ‖Lp(B(y,1)) <∞

for 1 ≤ p ≤ ∞ and B(y, 1) ⊂ RN being the unit balls centered at y. Following [7] we denote these spaces

by LpU

(R

N)

and consider their closed subspaces LpU

(R

N), of functions which are translation continuous with

respect to ‖ · ‖Lp

U

(RN), that is

‖τyφ− φ‖Lp

U

(RN) −→ 0 as |y| −→ 0,

where{τy, y ∈ R

N}

denotes the group of translations. Note in particular that L∞U

(R

N)

= L∞(R

N)

and

L∞U

(R

N)

= BUC(R

N). Similarly, we denote by W k,p

U

(R

N)

and W k,pU

(R

N), k ∈ N, Sobolev type spaces

constructed above LpU

(R

N)

and LpU

(R

N)

respectively. Thus, W k,pU

(R

N)

is the Banach space consisting of

elements φ ∈ W k,ploc

(R

N)

satisfying ‖φ‖W k,pU (RN ) =

∑|σ|≤k ‖Dσφ‖

LpU

(RN) < ∞ and W k,p

U

(R

N)

consists of

all those elements of W k,pU

(R

N), which are translation continuous with respect to ‖ · ‖W k,p

U (RN ). The case of

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com

Math. Nachr. 280, No. 15 (2007) 1645

non-integer k, W s,pU

(R

N), is defined by interpolation, see [7] and the Appendix for details. We just note here

that translations are continuous in these spaces.It is also shown in [7] that the differential operator A = ∆ considered in Lp

U

(R

N), 1 < p < ∞, with the

domain W 2,pU

(R

N), generates a C0 analytic semigroup eAt ∈ L(Lp

U

(R

N))

, t ≥ 0. Moreover, the correspondingfractional power spaces are the complex interpolation spaces and W 2α,p

U

(R

N)

:=[Lp

U

(R

N), W 2,p

U

(R

N)]

αfor

any 0 ≤ α ≤ 1.Therefore, using the techniques in [26], we can solve{

ut − ∆u = f(·, u,∇u), t > 0,

u(0) = u0 ∈ W 2α,pU

(R

N),

(1.6)

constructing the solution as a continuous function from [0, τu0) into W 2α,pU

(R

N)

satisfying the variation ofconstants formula

u(t, u0) = eAtu0 +∫ t

0

eA(t−s)f(·, u(s, u0),∇u(s, u0)) ds, t ∈ [0, τu0), (1.7)

provided f defines a locally Lipschitz map W 2α,pU

(R

N)→ Lp

U

(R

N).

Once local existence is established, the main idea that drives the arguments is to obtain suitable L∞(R

N)

bounds on the solutions. Then the variation of constants formula above allows us to transfer this estimate togradient estimates. Here is where the subquadratic growth assumption in the gradient is required for this techniqueto work. Finally note that when f depends on the gradient, L∞(

RN)

bounds are obtained by comparison witha suitable ODE (see e.g. (1.5)). On the other hand, if f depends only on x and u then comparison is carried outwith a suitable linear parabolic PDE.

Hence we first present the following local well possedness result for (1.6).

Proposition 1.1 i) Suppose that f : RN+1 → R, f = f(x, s), is locally Lipschitz continuous with respect to

s uniformly for x ∈ RN , and Holder continuous with respect to x uniformly for s varying in bounded subsets of

RN .Then, for p > N

2 and α ∈ [0, 1), such that 2α− Np > 0 and for each u0 ∈ W 2α,p

U

(R

N), there exists a unique

solution u = u(·, u0) to (1.6) defined on a maximal interval of existence [0, τu0).ii) Suppose that f : R

2N+1 → R, f = f(x, s,p), is a continuous function locally Lipschitz with respect to(s, p), uniformly for x ∈ R

N , and Holder continuous with respect to x uniformly for (s, p) varying in boundedsubsets of R

N+1.Then, for p > N and α ∈ [0, 1), such that 2α− N

p > 1 and for each u0 ∈ W 2α,pU

(R

N), there exists a unique

solution u = u(·, u0) to (1.6) defined on a maximal interval of existence [0, τu0).In either case, the solutions are given by (1.7) and satisfy

u ∈ C([0, τu0), W

2α,pU

(R

N)) ∩ C((0, τu0), W

2,pU

(R

N)) ∩C1

((0, τu0), W

2−,pU

(R

N))

(1.8)

where 2− denotes any positive number smaller than 2.

The above local well possedness result follows easily from the consideration of [26] and the fact that, in thefirst case of the theorem, W 2α,p

U

(R

N)

is embedded in BUC(R

N)

for 2α − Np > 0. For the second case the

argument is similar just noting that W 2α,pU

(R

N)

is embedded in C1bd

(R

N)

for 2α − Np > 1. In either case

the nonlinear term f defines a Nemytskiı map from W 2α,pU

(R

N)

to LpU

(R

N)

which is Lipschitz continuous onbounded sets.

Remark 1.2 Note that the assumptions on the nonlinearity in the proposition can be somewhat relaxed if oneassumes some particular structure. For example, when

f(x, s,p) = g(x) +m(x)s+ f0(x, s,p),

which will be of practical interests for the applications, it would suffice to assume g,m ∈ LpU

(R

N). Then the

proposition will hold provided that f0 satisfies the regularity stated in the proposition.

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To prove that local solutions in Proposition 1.1 are actually global in time, with bounded orbits, and to providedissipativeness we will require f to satisfy

sf(x, s, 0) ≤ 0, |s| ≥ s0 > 0, x ∈ RN , (1.9)

and

|f(x, s,p)| ≤ c(|s|)(1 + |p|γ0), (x, s,p) ∈ RN × R × R

N , (1.10)

where γ0 ∈ [1, 2) and c : [0,∞) → [0,∞) is a continuous function. These conditions are analogous as inthe case of semilinear equations in bounded domains (see [25, 36]). Note that once solutions are global thena C0-semigroup {T (t)} corresponding to (1.1) in W 2α,p

U

(R

N)

can be defined by T (t)u0 = u(t, u0), t ≥ 0,u0 ∈ W 2α,p

U

(R

N).

The main results of this article concerning solutions of (1.1) are formulated in Theorems 1.3, 1.5, and Corol-lary 1.4. Namely,

Theorem 1.3 Assume p > N and p ≥ 2 and f(x, s,p) as in Proposition 1.1. Assume also that (1.10) holdswith γ0 ∈ [1, 2) and (1.9) is satisfied.

Then we have the following resultsi) Problem (1.1) is globally well posed in W 2α,p

U

(R

N), 2α− N

p > 1. Orbits of bounded sets of the semigroup

{T (t)} of global solutions to (1.1) in W 2α,pU

(R

N), 2α− N

p > 1, are bounded and for each M > s0,

R ={φ ∈ W 2α,p

U

(R

N)

: ‖φ‖L∞(

RN) ≤M

}, (1.11)

is positively invariant set under {T (t)}.ii) If, in addition, (1.9) is strengthened to the condition

sf(x, s, 0) ≤ sh(s) < 0, |s| ≥ s0 > 0, x ∈ RN , (1.12)

for some continuous function h : R1 → R

1, then there exists a closed bounded invariant set A ⊂ W 2,pU

(R

N)

compact in W , which “attracts the asymptotic dynamics of (1.1)” in the sense that

distW (T (t)B,A) −→ 0, as t −→ ∞,

for any B bounded in W 2α,pU

(R

N). Here W denotes either C1+µ

loc

(R

N), with µ ∈ (0, 1 − N

p

), or the weighted

space W s,pρ

(R

N), 0 ≤ s < 2, where ρ is any translation of ρ0(x) =

(1 + |x|2)−ν

and ν > N2 .

If f does not depend on x, that is, f = f(s, p), then A is moreover invariant with respect to the group oftranslations in R

N .

Note that the meaning of attraction in W = C1+µloc

(R

N)

is that it takes place in C1+µ(B(0, R)) for eachR > 0. Also, note that the set A described in the Theorem 1.3 does not attract in the norm of the phase spaceW 2α,p

U

(R

N)

but the attraction takes place in a suitable weaker norm determined by the choices for the space W .Then we say that A is an

(W 2α,p

U

(R

N)−W

)attractor. Indeed, it is easy to see from the bistable example above

and the traveling wave solutions that since W 2α,pU

(R

N)↪→ L∞(

RN)

the set A cannot in fact attract in the normof the phase space W 2α,p

U

(R

N).

Also note that in particular when f does not depend on x (i.e., f = f(u,∇u)) we have, from (1.12),

Corollary 1.4 Assume f : RN+1 → R, f = f(s,p), is a locally Lipschitz function satisfying (1.10). Then,

for p > N , p ≥ 2, 2α − Np > 1, there exists a bounded

(W 2α,p

U

(R

N) − Lp

ρ0

(R

N))

attractor for (1.4) with

ρ0(x) =(1 + |x|2)−ν

and ν > N2 , if and only if the ordinary differential equation (1.5) has an attractor.

Evidently, in a non-gradient case f = f(u), assumption (1.10) in Corollary 1.4 above can be omitted.When the nonlinear term in (1.1) depends on x but does not depend on ∇u an even less restrictive structure

condition can be imposed on f . In this case, we will assume that

sf(x, s) ≤ C(x)|s|2 +D(x)|s|, s ∈ R, x ∈ RN , where

C,D ∈ LpU

(R

N)

for certain p >N

2, p ≥ 2 and D ≥ 0 a.e. in R

N ,(1.13)


Math. Nachr. 280, No. 15 (2007) 1647

see [6] for similar conditions in an Lp(R

N)

setting.We then have

Theorem 1.5 Suppose that f : RN+1 → R, f = f(x, s), is locally Lipschitz continuous with respect to s

uniformly for x ∈ RN , and Holder continuous with respect to x uniformly for s varying in bounded subsets of

RN . Suppose also that (1.13) holds and the linear semigroup generated by ∆ +CI decays exponentially, that is,

Reσ(−∆ − C(·)I) > 0.Then for p > N

2 and α ∈ [0, 1) such that 2α − Np > 0 there exists a closed bounded invariant set

A ⊂ W 2,pU

(R

N), which is a

(W 2α,p

U

(R

N) −W

)attractor for {T (t)}, compact in W , where W denotes ei-

ther Cµloc

(R

N), with 2− N

p > µ > 0, or the weighted spaceW s,pρ

(R

N), 0 ≤ s < 2, with ρ being any translation

of ρ0(x) =(1 + |x|2)−ν

, ν > N2 .

Note that the results in [7] allow us to consider the operator A = ∆ (and in fact much more general ellipticoperators) in the larger space Lp

U

(R

N)

or even in the much larger space Lpρ

(R

N). However in the former case

the operator has non dense domain W 2,pU

(R

N)

and therefore the linear semigroup is not continuous at t = 0.The nonlinear equation could then be handled using the techniques for non densely defined generators as in [31].Note that this type of difficulties is the same as one finds in solving the heat equation in a bounded domain withDirichlet boundary conditions or in R

N , with L∞ as a phase space. This is also the reason why the BUC settingis somewhat easier for local existence than the L∞ setting, see e.g. [32].

On the other hand for the case of weighted spaces, although the linear semigroup is continuous (see [7]),the difficulties connected with the nonlinear equations come from the lack of Sobolev-like embeddings whichmake unavailable the standard techniques for handling nonlinear terms. In fact it was shown in [8] that generally,without specific structure assumption on nonlinear terms, semilinear problems are ill posed in weighted spaceswhen the weight goes to zero at infinity. However with monotonicity assumptions of the form

∂f

∂s(x, s) ≤ const, x ∈ R

N , s ∈ R,

the nonlinear semigroups can be constructed, see e.g. [13]. See also [35] for even less restrictive monotonicityconditions or suitable growth and structure conditions in weighted spaces.

Finally, note that we could have considered (1.1) in less regular spaces, e.g. without assuming p > N orp > N/2 in Proposition 1.1, but assuming some growth conditions on the nonlinearities, as in [6] for the Lp

(R

N)

setting. However, for simplicity we will not pursue along this direction here.The main results above will be proved in Section 2. In Section 3 we will discuss several examples and compare

the structure conditions appearing in the references with the ones in this paper. Finally in the Appendix we willgive suitable characterization of the spaces W s,p

U

(R

N)

which are used in previous sections. Such a result willsupplement earlier consideration of [7].

2 Proofs of the main results

Before proving the main results of this paper we recall several useful properties of the spaces W s,pU

(R

N)

andW s,p

ρ

(R

N)

that have been proved in [7]; see also the Appendix. Among them we mention thatC∞bd

(R

N)

is dense

in W s,pU

(R

N)

and if ρ is any translation of ρ0(x) =(1 + |x|2)−ν

with ν > N2 , then we have the continuous

embeddings

W s,pU

(R

N)↪→ W s,p

ρ

(R

N), (2.1)

and

W s,pU

(R

N)↪→ Cl+µ

(R

N), l + 1 > s− N

p> l + µ > l, (2.2)

while the embeddings

W 2,pU

(R

N)↪→W 1,p

ρ

(R

N)

(2.3)



and

W s,pU

(R

N)↪→ Cl+µ(Q), l + 1 > s− N

p> l + µ > l, (2.4)

are compact, where Q is any compact set in RN . See [7, Lemma 4.1] and [12].

Now, we note that the conclusions of Theorems 1.3 and 1.5 will be a consequence of the following generalresults.

Proposition 2.1 Assume {S(t)} is a C0-semigroup on a metric space V for which there exists a bounded setB0 ⊂ V absorbing bounded sets and W is a metric space containingB0 such that

any sequence {S(tn)vn} where tn −→ ∞ and {vn} ⊂ B0

possesses a subsequence convergent in the metric of W .(2.5)

Then there exists a nonempty set A ⊂ W which is compact in W and attracts bounded subsets of V withrespect to the Hausdorff semidistance dW .

P r o o f. The assertion follows from (2.5), defining A as

A :={w ∈W : S(tn)vn

W→ w for certain {vn} ⊂ B0 and tn → ∞},

and the property that B0 is an absorbing set.

Note that the set A defined in the proposition above is a minimal element in the class of compact subsets ofW attracting bounded subsets of V in the metric of W and, if W = V , it is a global attractor in the sense of [24].

The set A resulting from Proposition 2.1 will be called a (V −W ) attractor for {S(t)} if additionally A isalso a closed subset of V , invariant under {S(t)}.

For example (2.5) holds whenever V is compactly embedded in W and in this case A ⊂ W . However, ingeneral, one has to impose conditions additional to (2.5) to conclude that A is contained in V and is an invariantset. For example, if V is a reflexive Banach space (or more generally the dual of a Banach space), then (2.5)together with weak (or weak-∗) convergence, gives that A ⊂ V .

Motivated by this we extend the both notions of asymptotic compactness [29] and asymptotic smoothness [24]introducing the following ‘generalized asymptotic smoothness condition’.

For any tn −→ ∞ and {vn} ⊂ B0, {S(tn)vn} contains a subsequence

{S(tnk)vnk

} convergent in the metric of W to certain v ∈ V ∩W and,

if in addition, S(tnk+ t)vnk

W−−→ v for certain t > 0, then S(t)v = v.

(2.6)

Note that the idea of using ω-limit sets, in weaker topologies, of absorbing sets to construct a suitable notionof attractor has been used before, see e.g. [9], where the weak topology is considered in an L2

(R

N)

setting.

Corollary 2.2 Suppose that assumptions of Proposition 2.1 hold and, in addition, (2.6) is satisfied and A isclosed in V . Then {S(t)} possesses a (V −W ) attractor A.

We remark that in applications we present in this paper, specific properties of equations of the form (1.1) willallow us to prove that (V −W ) attractor A is in fact bounded in V , as stated in Theorems 1.3 and 1.5.

Now we derive necessary estimates and prove the results reported in Section 1. The advantage of the functionalanalytic setting in uniformly local spaces is that most estimates will follow as for equations in bounded domains.

2.1 Proof of Theorem 1.3

In this subsection we consider (1.1) with f : R2N+1 → R, f = f(x, s,p), locally Lipschitz with respect to (s, p)

uniformly for x ∈ RN , and Holder continuous with respect to x uniformly for (s, p) varying in bounded subsets

of RN+1. As follows from Proposition 1.1, if p > N , α ∈ [0, 1) and 2α − N

p > 1, then (1.1) is locally well

posed in W 2α,pU

(R

N). Under further assumptions of Theorem 1.3 we prove below that the solutions to (1.1) exist

globally in time and the corresponding semigroup in W 2α,pU

(R

N)

is bounded dissipative. We then show that the


Math. Nachr. 280, No. 15 (2007) 1649

generalized asymptotic smoothness condition (2.6) is satisfied and construct a(W 2α,p

U

(R

N),W)

attractor, as inCorollary 2.2.

This will be done in a sequence of lemmas below.

Lemma 2.3 Suppose that f : R2N+1 → R, f = f(x, s,p), p = (p1, . . . , pN ) is locally Lipschitz continuous

with respect to s and p uniformly for x ∈ RN as well as it is µ-Holder continuous with respect to x, uniformly

for (s, p) varying in bounded subsets of R × RN . Suppose also that (1.10) holds with γ0 ∈ [1, 2] and in addition

sf(x, s,0) ≤ b1s2 + b2 for (x, s) ∈ R

n × R1, where b1, b2 are positive constants.

If v = v(t, x), v = v(t, x) ∈ C1+ µ2 ,2+µ

([0, T ]× R

N)

are such that

vt − ∆v − f(x, v,∇v) ≤ 0 ≤ vt − ∆v − f(x, v,∇v), t ∈ [0, T ], x ∈ RN ,

and

v(0, x) = v0(x) ≤ v(0, x) = v0(x), x ∈ RN ,

then

v ≤ v in [0, T ]× RN .

P r o o f. If we define the function r(t, x) := vt−∆v−f(x, v,∇v), then v becomes a uniqueC1+ µ2 ,2+µ

([0, T ]×

RN)-solution of the Cauchy problem{

vt − ∆v − f(x, v,∇v) = r(t, x), t ∈ (0, T ], x ∈ RN ,

v(0, x) = v0(x), x ∈ RN .

Following [30, Chapter V, §8], v can be then approximated pointwise in [0, T ] × RN by a sequence {vn} of

classical solutions to the Dirichlet initial boundary value problems⎧⎪⎪⎨⎪⎪⎩vnt − ∆vn − f(x, vn,∇vn) = r(t, x), t ∈ (0, T ], x ∈ Bn,

vn(t, x) = v0(x), x ∈ ∂Bn, t ∈ (0, T ],

vn(0, x) = v0(x), x ∈ Bn,

(2.7)

defined in an increasing sequence of cylinders (0, T ] × Bn, where Bn ={x ∈ R

N : |x| < n}

and vn isbounded in Bn together with all its first order partial derivatives with respect to the space variables. Since vcan be approximated similarily by a correspondig sequence {vn}, the assertion follows as a consequence ofthe Nagumo–Westphal type comparison result (see [44, Chapter IV, §24]) applied to the Dirichlet problems inBn.

Lemma 2.4 If (1.10) holds with γ0 ∈ [1, 2) and (1.9) is satisfied then the solutions to (1.1) resulting fromProposition 1.1 are bounded inL∞(

RN)

and in W 2α,pU

(R

N)

uniformly for t ∈ [0,∞) and u0 varying in bounded

subsets of W 2α,pU

(R

N).

P r o o f. Since the constants ±M where M > max{‖u0‖L∞(RN ), s0

}are respectively supersolution and

subsolution to (1.1), Lemma 2.3 i) ensures that local solutions to (1.1) through u0 ∈ C∞bd

(R

N)

are bounded apriori in L∞(

RN)

uniformly for t ∈ [0,∞) and for u0 varying in bounded subsets of W 2α,pU

(R

N). The same

property holds for the solutions through u0 ∈ W 2α,pU

(R

N)

becauseC∞bd

(R

N)

is dense in W 2α,pU

(R

N)

and hence

supt∈[0,∞)

sup‖u0‖W

2α,pU

(RN )≤R

‖u(t, u0)‖L∞(

RN) ≤ c(R). (2.8)

We next observe that (1.10) with γ0 ∈ [1, 2) leads to the estimate

‖f(x, u,∇u)‖LpU(RN ) ≤ g

(‖u‖L∞(RN )

) (1 + ‖u‖θ

W 2γ,pU (RN )

), (2.9)



where max{α, γ0

2

}< γ < 1, θ ∈ ( 1

2γ ,1γ0

), and g : [0,∞) → [0,∞) is a continuous function. There exists also

τR > 0 such that ‖u(t, u0)‖W 2α,pU (RN ) is uniformly bounded with respect to t ∈ [0, τR], ‖u0‖W 2α,p

U (RN ) ≤ R; inparticular,

V ={v : ∃ ‖u0‖W 2α,p

U (RN ) ≤ R, v = u(τR, u0)}

is a bounded subset of W 2γ,pU

(R

N)

(2.10)

(see [14, Proposition 2.3.1, 3.2.1]).Using (2.9), (2.10) and a modification of the variation of constants formula (1.7) we obtain

‖u(t, v)‖W 2γ,pU (RN )

≤ ∥∥e(∆−ωI)t∥∥L(Lp

U (RN ),LpU (RN ))‖v‖W 2γ,p

U (RN )

+∫ t

0

∥∥e(∆−ωI)(t−s)∥∥L(Lp

U (RN ),W 2γ,pU (RN ))g

(‖u(s, v)‖L∞(RN )

)(1 + ‖u(s, v)‖θ

W 2γ,pU (RN )

)ds

+∫ t

0


U (RN ),W 2γ,pU (RN ))ω‖u(s, v)‖Lp

U(RN ) ds.

(2.11)

We chose ω ∈ R in (2.11) in such a way that Re σ(−∆ + ωI) > a > 0 and then


U (RN ),W 2β,pU (RN )) ≤ cβ

e−at

tβ, t > 0, β ∈ [0, 1]. (2.12)

Also ‖u(s, v)‖LpU(RN ) does not exceed a multiple of ‖u(s, v)‖L∞(RN ), which has already been estimated in (2.8).

This allow us to get from (2.11) the inequality

sups∈[0,t]

supv∈V

‖u(s, v)‖W 2γ,pU (RN )

≤ c0 supv∈V

‖v‖W 2γ,pU (RN )

+ sups∈[0,C(R)]

(g(s) + ωs)

(1 + sup

s∈[0,t]

supv∈V

‖u(s, v)‖θW 2γ,p

U (RN )

)∫ ∞

0

cγe−ay

yγdy,

and observe that z := sups∈[0,t] supv∈V ‖u(t, v)‖W 2γ,pU (RN ) satisfies the relation

0 ≤ z ≤ const(1 + zθ

). (2.13)

Note that the value of const essentially depends on R and that any number satisfying (2.13) is bounded above bythe positive root z0 = c(R) of the equation z = const

(1 + zθ

). The latter ensures the estimate

supt∈[0,∞)

supv∈V

‖u(t, v)‖W 2γ,pU (RN ) ≤ c(R). (2.14)

Having both (2.14) and the boundedness of ‖u(t, u0)‖W 2α,pU (RN ) uniform with respect to t ∈ [0, τR] and

‖u0‖W 2α,pU (RN ) ≤ R, it is next routine to get

supt∈[0,∞)

sup‖u0‖

W2α,pU

(RN )≤R

‖u(t, u0)‖W 2α,pU (RN ) ≤ const(R), (2.15)

which is the desired bound on the solutions.

Remark 2.5 As a consequence of Lemma 2.4 the solutions to (1.1) in W 2α,pU

(R

N)

exist globally in time andorbits of bounded sets of the semigroup {T (t)} corresponding to (1.1) in W 2α,p

U

(R

N)

are bounded. Recalling thatconstant functions ±M satisfying M > max

{‖u0‖L∞(RN ), s0}

are respectively supersolution and subsolutionto (1.1) it is clear that the set R defined in (1.11) is positively invariant. Part i) of Theorem 1.3 is thus proved.


Math. Nachr. 280, No. 15 (2007) 1651

Lemma 2.6 If, in addition to assumptions of Lemma 2.4, condition (1.12) holds, then the L∞(R

N)

and

W 2α,pU

(R

N), 2α − N

p > 1, estimates (2.8), (2.14) are asymptotically independent of u0 varying in bounded

subsets of W 2α,pU

(R

N).

P r o o f. Note that, from (1.12), considering the solutions of the ODE

U = h(U), t > 0, (2.16)

with initial condition U(0) = s ∈ R, if u0 belongs to a ball in W 2α,pU

(R

N)

of radius R > 0 and M > s0is chosen sufficiently large then, by Lemma 2.3 and density of C∞

bd

(R

N)

in W 2α,pU

(R

N), u(t, u0) will remain

between the solutions U(t,−M) and U(t,M) to (2.16). Evidently U(t,±M) approaches appropriate zeros ofh(s) as t → ∞, which translates into the estimate

lim supt→∞

sup‖u0‖

W2α,pU

(RN )≤R

‖u(t, u0)‖L∞(RN ) ≤ s0. (2.17)

The above relation ensures that if R > 0 is arbitrarily fixed, there exists tR > 0 such that any solution with‖u0‖W 2α,p

U (RN ) ≤ R remains bounded by s0 uniformly for t > tR and x ∈ RN . From (2.11), (2.8), (2.14), (2.17)

we then have for τ > tR


≤ ∥∥e(∆−ωI)t∥∥L(Lp

U (RN ),LpU (RN ))‖v‖W 2γ,p

U (RN )

+∫ τ

0


U (RN ),W 2γ,pU (RN )) sup

s∈[0,C(R)]

(g(s) + ωs)(1 +[c(R)

]θ)ds

+∫ t

τ


U (RN ),W 2γ,pU (RN )) sup

s∈[0,s0]

(g(s) + ωs)(1 + ‖u(s, v)‖θ

W 2γ,pU (RN )

)ds

and consequently, using (2.12),

lim supt→∞

supv∈V


≤ sups∈[0,s0]

(g(s) + ωs)

(1 + sup

s∈[τ,∞]

supv∈V

‖u(s, v)‖θW 2γ,p

U (RN )

)∫ ∞

0

cγe−ay

yγdy, τ > tR.

The above inequality ensures that lim supt→∞ supv∈V ‖u(t, v)‖W 2γ,pU (RN ) satisfies the relation like (2.13) but

now with constant independent of R > 0. We thus obtain

lim supt→∞

supv∈V

‖u(t, v)‖W 2γ,pU (RN ) ≤ z0, (2.18)

where z0 is the positive root of the equation

z =

[sup

s∈[0,s0]

(g(s) + ωs)∫ ∞

0

cγe−ay

yγdy

](1 + zθ

)and thus z0 does not depend on R > 0.

After establishing asymptotic bounds on the solutions the next result shows that condition (2.5) is satisfied.The more difficult part is, of course, proving that the limit points are in fact in W 2α,p

U

(R

N).

Lemma 2.7 Under the assumptions of Lemma 2.6 the semigroup {T (t)} of global solutions to (1.1) inW 2α,p

U

(R

N)

possesses an absorbing set B0, bounded in W 2,pU

(R

N)

and positively invariant under {T (t)}.Furthermore, for any sequence of the form {T (tn)un}, where tn → ∞ and {un} ⊂ B0, there exits a

subsequence that converges inW to certainw ∈ W 2α,pU

(R

N), whereW denotes eitherC1+µ

loc

(R

N), with 1−N

p >

µ > 0, or the weighted space W s,pρ

(R

N), 0 ≤ s < 2, where ρ is any translation of ρ0(x) =

(1 + |x|2)−ν

andν > N

2 .



P r o o f. It has already been mentioned in Remark 2.5 that (1.1) generates on W 2α,pU

(R

N)

a C0 semigroup{T (t)} with bounded orbits of bounded sets. Estimate (2.18) ensures now that {T (t)} has a bounded absorbingset B0 ⊂ W 2α,p

U

(R

N). The positive orbit γ+

(B0

)of the absorbing set B0 is also bounded in W 2α,p

U

(R

N).

Moreover, as a result of [14, Lemma 3.2.1], the set T (1)γ+(B0

)is in fact bounded in W 2,p

U

(R

N)

so that the first

assertion of the lemma holds with B0 := T (1)γ+(B0

).

For the proof of the second assertion consider a sequence {T (tn)un}, where tn → ∞ and {un} ⊂ B0.Using the compact embeddings (2.3) and (2.4) we choose a subsequence {T (tnk

)unk}, which is convergent in

W 1,pρ

(R

N)

and inC1+µloc

(R

N), to a functionw in these spaces. Hence, it remains to prove thatw ∈ W 2α,p

U

(R

N).

Since {T (tnk)unk

} ⊂ B0 is bounded in W 2,pU

(R

N), then in particular we have that

‖T (tnk)unk

‖W 2,p(B(y,1)) ≤ C

for every k and y ∈ RN . Then by weak convergence in W 2,p(B(y, 1)) we get that ‖w‖W 2,p(B(y,1)) ≤ C for

every y ∈ RN and then w ∈ W 2,p

U

(R

N)↪→ W 2α,p

U

(R

N), as will be shown in the Appendix.

Remark 2.8 From the proof of Lemma 2.7, we obtain that under the assumptions of Lemma 2.6 there existsCB0 > 0 such that if w is a limit in W s,p

ρ

(R

N)

with certain s ∈ [0, 2) of a sequence {T (tn)un}, where tn → ∞and {un} ⊂ B0, then

‖w‖W 2,p

U

(RN) ≤ CB0 .

Hence the set A in Proposition 2.1 is actually bounded in W 2,pU

(R

N)↪→ W 2α,p

U

(R

N). Therefore, as a conse-

quence of [14, Lemma 3.2.1], A is in fact bounded in W 2,pU

(R

N).

Now, in the next two lemmas, we show that the generalized asymptotic smoothness condition (2.6) is satisfied.

Lemma 2.9 Under the assumptions of Lemma 2.6, if γ ∈ [0, 1), {un} ⊂ B0, u0 ∈ W 2α,pU

(R

N) ∩

W 2γ,pρ

(R

N)

and {un} tends to u0 in W 2γ,pρ

(R

N), then {u(t, un)} converges to u(t, u0) in W 2γ,p

ρ

(R

N)

foreach t > 0.

P r o o f. As shown in [7], the operator A considered in Lpρ

(R

N)

with the domain W 2,pρ

(R

N)

is sectorialin Lp

ρ

(R

N), 1 < p < ∞. Furthermore, fractional power spaces associated to A in Lp

ρ

(R

N)

coincide with thecomplex interpolation spaces, (see the Appendix)

W 2β,pρ

(R

N)

=[Lp

ρ

(R

N),W 2,p

ρ

(R

N)]

β. (2.19)

If Ff (φ)(x) := f(x, φ(x),∇φ(x)), φ ∈ W 2,pU

(R

N)

and B0 is an invariant absorbing set from Lemma 2.7then

‖Ff(v) −Ff (w)‖Lp

ρ

(RN) ≤ L ‖v − w‖

W 2β,pρ

(RN), v, w ∈ B0, β ∈ [ 12 , 1), (2.20)

because f is locally Lipschitz continuous function and supφ∈B0max|σ|≤1 |Dσφ(x)| is bounded by a multiple

of supφ∈B0‖φ‖W 2,p

U (RN ). Also, if {un} ⊂ B0 is convergent to u0 in W 2γ,pρ

(R

N)

for certain γ ∈ [0, 1], then

Lemma 2.7 ensures that in fact u0 belongs to W 2γ,pρ

(R

N)

with any γ ∈ [0, 1] and {un} is convergent to u0 inthe latter spaces as well. It is thus clear that it suffices to prove the assertion for γ close to 1.

Thanks to (2.20), for the solutions u(t, un), u(t, u0) to (1.1) and each γ ∈ [12 , 1), 0 < t < τ <∞ we have

‖u(t, un) − u(t, u0)‖W 2γ,pρ (RN )

≤ ∥∥e∆t∥∥L(Lp

ρ(RN ),Lpρ(RN ))

‖un − u0‖W 2γ,pρ (RN )

+∫ t

0

∥∥e∆(t−s)∥∥L(Lp

ρ(RN ),W 2γ,pρ (RN ))

‖Ff(u(s, un)) −Ff(u(s, u0))‖Lpρ(RN ) ds

≤ c(τ)‖un − u0‖W 2γ,pρ (RN ) + Lc′(τ)

∫ t

0

(t− s)−γ‖u(s, un) − u(s, u0)‖W 2γ,pρ (RN ) ds.


Math. Nachr. 280, No. 15 (2007) 1653

Application of the singular Gronwall lemma (see [26]) allows us to conclude that

‖u(t, un) − u(t, u0)‖W 2γ,p

ρ

(RN) −→ 0 as n −→ ∞

for arbitrarily chosen t > 0.

Lemma 2.10 Suppose that the assumptions of Lemma 2.6 hold.If tn → ∞ and {un} ⊂ B0, then {T (tn)un} contains a subsequence {T (tnk

)unk} convergent inC1+µ

loc

(R

N),

µ ∈ (0, 1 − Np

), to certain v ∈ W 2α,p

U

(R

N)

and, in addition,

T (tnk+ t)unk

C1+µloc (R

N)−−−−−−−→ T (t)v for each t > 0.

P r o o f. Let tn → ∞ and {un} ⊂ B0. From Lemma 2.7 there exists {T (tnk)unk

} and v ∈ W 2α,pU

(R

N)

such that {T (tnk)unk

} converges to v both in C1+µloc

(R

N), µ ∈ (0, 1− N

p

), and in W s,p

ρ

(R

N)s ∈ [0, 2). Hence

from Lemma 2.9, we get

T (tnk+ t)unk

W s,pρ (R

N)−−−−−−−→ T (t)v for each t > 0. (2.21)

Since we can apply Lemma 2.7 to arbitrarily chosen subsequence of {T (tnk+ t)unk

}, we get from (2.21) that

T (tnk+ t)unk

C1+µloc (R

N)−−−−−−−→ T (t)v

for each t > 0, µ ∈ (0, 1 − Np

).

Hence (2.6) is satisfied and Corollary 2.2 ensures the existence of a(W 2α,p

U

(R

N) −W

)attractor for (1.1),

where W is either a Holder space C1+µloc

(R

N), µ ∈ (0, 1 − N

p

), or a weighted space W s,p

ρ

(R

N), s ∈ [0, 2).

These attractors are independent of the choice of W and can be described as

A :={w ∈ W 2α,p

U

(R

N)

: T (tn)unW→ w for certain {un} ⊂ B0 and tn → ∞

},

see Remark 2.8.To complete the proof of Theorem 1.3 we show that when f does not depend on x, then A is invariant with

respect to the group of translations in RN .

Lemma 2.11 Suppose that the assumptions of Lemma 2.6 hold and assume moreover that f does not dependon x, that is, f = f(s, p). Then

w(·) ∈ A if and only if w(· − z) ∈ A for each z ∈ RN .

P r o o f. Note that if φ ∈ W 2,pU

(R

N)

then φ(· − z) ∈ W 2,pU

(R

N)

for each z ∈ RN and recall that B0 is

positively invariant absorbing set for {T (t)}, which is also bounded in W 2,pU

(R

N). Fix z ∈ R

N , w ∈ A andtn → ∞, {un} ⊂ B0 such that T (tn)un → w in W being either a Holder space C1+µ

loc

(R

N), µ ∈ (0, 1 − N

p

),

or a weighted space W s,pρ

(R

N), s ∈ [0, 2). In particular T (tn)un → w point-wise in R

N . Then w(· − z) ∈W 2,p

U

(R

N), the sequence {un(· − z)} is bounded in W 2,p

U

(R

N)

and consequently T (t){un(· − z)} ⊂ B0 fort ≥ t0.

Define further

tn = tn0+n − t0, un = T (t0)un0+n(· − z), n ∈ N.

Note that tn → ∞, {un} ⊂ B0 and consider T (tn){un}.By Lemma 2.7 there is a subsequence {T (tnk

)unk} convergent to a certain w ∈ A in W . In particular the

convergence is point-wise in RN . Since T (tnk

)unk(·) = T (tn0+nk

)un0+nk(· − z) and T (tn0+nk

)un0+nk(x) →

w(x) for x ∈ RN we obtain that T (tnk

)unk(x) → w(x − z) for x ∈ R

N . This ensures that w(·) = w(· − z)and thus w(· − z) ∈ A.



Remark 2.12 Note that in the situation above translations in RN are symmetries of the equation, in the sense

that for all t ≥ 0 and y ∈ RN we have

τyT (t) = T (t)τy.

In this context the result in Lemma 2.11 can be generalized in the following way: if g is a bounded lineartransformation in W 2α,p

U

(R

N)

such that

g ◦ T (t) = T (t) ◦ gthen g(A) = A; see [37, Propositions 1.2 and 1.3].

2.2 Proof of Corollary 1.4

If (1.5) has an attractor in R, then sf(s, 0) < 0 for large |s|. Then the “if” part follows from Theorem 1.3 withh(s) = f(s, 0).

On the other hand, assume that (1.4) has a bounded(W 2α,p

U

(R

N) − Lp

ρ0

(R

N))

-attractor A, where ρ0(x) =(1 + |x|2)−ν

with ν > N/2. In particular, if B(rA) is a ball in Lpρ0

(R

N)

centered at zero and containing A,

then the orbit of any bounded set B ⊂ W 2α,pU

(R

N)

must enter B(rA) in certain time tB ≥ 0 and remain insidethis ball for all t ≥ tB . Note that this happens as well with the solutions U(·, U0) of (1.5) because they also solve(1.4).

Solutions U(·, U0) are thus defined globally in time and, moreover,

lim supt→+∞

‖U(t, U0)‖Lp

ρ0

(RN) = lim sup

t→+∞|U(t, U0)|

(∫RN

ρ0 dx

) 1p

≤ rA.

Hence (1.5) has an attractor.

2.3 Proof of Theorem 1.5

In this subsection we discuss the Cauchy problem for

ut = ∆u+ f(x, u), x ∈ RN , t > 0, (2.22)

where the function f : RN+1 → R is locally Lipschitz continuous with respect to u uniformly for x ∈ R

N ,and Holder continuous with respect to x uniformly for u varying in bounded subsets of R

N . We consider u0 ∈W 2α,p

U

(R

N)

with α ∈ [0, 1), p > N2 , p ≥ 2, 2α − N

p > 0 and use comparison techniques (see e.g. [5,Appendix A]) to derive suitable estimates of the solutions.

Lemma 2.13 Under the above assumptions and (1.13) solutions u(t, u0) to (2.22) are bounded in L∞(R

N)

on any compact time interval uniformly with respect to initial condition u0 varying in bounded subsets ofW 2α,p

U

(R

N).

P r o o f. Note that (1.13) implies the relations

f(x, s) ≤ C(x)s +D(x), s ≥ 0, x ∈ RN ,

f(x, s) ≥ C(x)s −D(x), s ≤ 0, x ∈ RN .

Since the resolvent (λI − ∆)−1 of ∆ in Lp(R

N)

is increasing for large λ, the heat semigroup {e∆t} is orderpreserving in locally uniform spaces, (see [7, Proposition 5.22 and Remark 5.11]). Therefore, by monotonicityand comparison with the solutions of a linear problem in Lp

U

(R

N)

Ut = ∆U + C(x)U +D(x), U(0) = |u0|, (2.23)

we observe that |u(t, u0)| must stay below U(t, |u0|). The latter function is bounded in W 2α,pU

(R

N)

on compactsubintervals of (0,∞) and uniformly for ‖u0‖W 2α,p

U (RN ) ≤ R. We also remark that for any R > 0 there exists


Math. Nachr. 280, No. 15 (2007) 1655

τR > 0 such that solutions u(·, u0) to (2.22) with ‖u0‖W 2α,pU (RN ) ≤ R are bounded in W 2α,p

U

(R

N)

uniformly

for t ∈ [0, τR] and ‖u0‖W 2α,pU (RN ) ≤ R (see [14, Proposition 2.3.1]). Since under our assumptions on α and p

we have that W 2α,pU

(R

N)↪→ L∞(

RN), the required L∞(

RN)

estimate of u(·, u0) is straightforward.

Lemma 2.14 If additionallyRe(−∆−C(·)I) > 0, then the L∞(R

N)

estimate obtained in Lemma 2.13 canbe strengthened from compact time intervals to the whole half line [0,∞) and, moreover, it remains asymptoticallyindependent of initial condition u0 varying in bounded subsets of W 2α,p

U

(R

N).

P r o o f. Since Re σ(−∆ − C(·)I) > 0, the solution to (2.23) can be decomposed as

U = V + Φ,

where

Φ = (−∆ − C(·)I)−1D ∈ W 2,pU

(R

N)

(2.24)

and V solves the linear equation

Vt = ∆V + C(x)V, t > 0, x ∈ RN .

We thus have that

U = e(∆+C(·)I)t(U(0) − Φ) + Φ, (2.25)

and, recalling that the semigroup{e(∆+C(·)I)t

}is exponentially decaying, we obtain the estimate of U in

W 2α,pU

(R

N)↪→ L∞(

RN)

uniform on infinite time intervals away from t = 0. The latter estimate is nexttranslated into L∞(

RN)

bound on u(·, u0) based on comparison methods. As a consequence we obtain both

supt≥0

sup‖u0‖W

2α,pU

(RN )≤R

‖u(t, u0)‖L∞(RN ) ≤ C(R) (2.26)

and

lim supt→∞

sup‖u0‖W

2α,pU

(RN )≤R

‖u(t, u0)‖L∞(RN ) ≤ ‖Φ‖L∞(RN ). (2.27)

The remaining part of the proof of Theorem 1.5 is similar as in the case of Theorem 1.3. In particular, theexistence of an attractor follows as before from Corollary 2.2.

3 Comparison with known results and examples

In the last fifteen years a number of problems in unbounded domains originating in applications have been in-tensively studied [1, 6, 9, 12, 13, 14, 15, 16, 18, 19, 20, 23, 32, 33, 34, 45]. The references contain merely asample list of the related publications. Observe that in none of these references the existence of an attractor for ascalar semilinear parabolic equation in the whole of R

N is obtained, as in this paper, under the same assumptionsas employed for problems in bounded domains. Also in contrary to the method used in a number of referencesdevoted to the Cauchy problems in R

N , like e.g. [9, 18, 19, 45], a functional analytic setting and a consequentanalytic semigroup approach in locally uniform spaces constitute in the present paper a main tool both to obtainlocal well posedness result and to derive sufficiently smooth estimates of the solutions. Such an approach is usedin the monographs [26, 2, 31, 14] and the present paper remains in the same vein.

Very recently, attractors for semilinear parabolic equations and systems have been investigated in [13, 18, 19]and [45].

Here we compare several known structure conditions on nonlinear terms with the ones in this paper.



Example 3.1 As for the gradient dependent nonlinearities f = f(x, u,∇u) in scalar equations the main“structure” condition of [18, 19] reads:

f(x, u,∇u) = −λ0u+ f0(u,∇u) + g(x), where uf0(u,∇u) ≤ c and λ0 > 0, (3.1)

plus the subquadratic growth restriction concerning ∇u. In [18] a polynomial bound of |f | with respect to u aswell as a strong “sign” condition uf0(u,∇u) ≤ 0 were also required. The latter two requirements were weakenedin [19], however, in both mentioned publications some assumptions on the derivatives f ′

u, f ′∇,u are used.

The results of the present paper allow us to consider nonlinearities for which (3.1) is not satisfied, e.g.

f(x, u,∇u) = a(x, u)|∇u|γ0 + b(x, u),

with 0 ≤ γ0 < 2, a(x, s), b(x, s) locally Lipschitz with respect to s uniformly for x ∈ RN and Holder continous

with respect to x uniformly for s varying in bounded subsets of R and such that sb(x, s) ≤ sh(s) < 0 for|s| > s0 > 0, x ∈ R

N .In particular the Burgers equation

ut = uxx + uux, x ∈ R, t > 0, (3.2)

is an important example for which (3.1) fails although Theorem 1.3 i) applies.Even if a stabilizing and a suitable source term are added in (3.2) so that we have

ut = uxx − λ0u+ uux + g(x), x ∈ R, t > 0, (3.3)

condition (3.1) of [18] fails again, whereas the applicability of Theorem 1.3 ii) is straightforward.

Example 3.2 For the non-gradient case, the assumptions in [45] come from the earlier publication [9] andread: f = f(u) and

f(u) = −λ0u+ f0(u) where λ0 > 0,

uf0(u) ≤ c, f ′0(u) < c,

|f(u)| ≤ C(1 + |u|p) with p < p0 =N

N − 4if N > 4 or p arbitrarily large if N ≤ 4.

(3.4)

Note that in [9] the maximal exponent p0 could not exceed min{

4N ,

2N−2

}for N ≥ 3, in which case a nonlin-

earity like u(1−|u|r−1

)in space dimensionsN ≥ 3 requires a severe growth restriction. In [13] the nonlinearity

f = f(u) was much like in (3.4) above but the growth exponent p could be arbitrarily large independently of thespace dimensionN .

Conditions (3.4) imply uf(u) ≤ −λ0u2 + c while, as shown in the present paper, a necessary and sufficient

condition for the existence of an attractor is in fact the much weaker condition uf(u) < 0 for large |s| (seeCorollary 1.4).

Dissipativeness assumption (1.13) appearing in Theorem 1.5 has already been exploited in [6] in the discussionof attractors for reaction-diffusion equations in “standard” Lebesgue spaces. In the present (locally uniformspace) setting the mentioned assumption becomes even more flexible in applications. Also in some specialsituations it is possible to observe that the solutions enter asymptotically a “smaller” space consisting of integrablefunctions. Observe that it is easy to see, e.g. considering a nonzero constant initial data in the heat equation, thatthis latter property cannot be true in general.

Example 3.3 Recall that a sample nonlinearity

f(x, s) = m(x)s− s3, (3.5)

has been studied in [32] in a BUC(R

N)

setting under the requirement that the functionm ∈ Cµ(R

N)

was a dif-

ference of the continuous mapsm+ andm−, m+ ∈ C0

(R

N)

andm− is such that{e(∆−m−)t

}is asymptotically

decaying.


Math. Nachr. 280, No. 15 (2007) 1657

In [6] the same model example was considered in standard Lq(R

N)

spaces. It was then shown that theequation is dissipative in that space under the assumption that m ∈ Lp

U

(R

N), p > N

2 , p ≥ 2, the semigroup{e(∆−m−)t

}decays exponentially and

limR→∞

‖m+‖LpU (RN\B(0,R)) = 0.

With the techniques of this paper, note that the assumptions in Theorem 1.5 are satisfied only assuming m ∈Lp

U

(R

N), p > N

2 , p ≥ 2. Indeed, the embedding W 2α,pU

(R

N)↪→ L∞(

RN), 2α > N

p , ensures that the

corresponding Nemytskiı map takes W 2α,pU

(R

N)

into LpU

(R

N)

and is Lipschitz continuous on bounded sets.On the other hand, note that (1.13) is satisfied with C equal to a negative constant and D(x) being a multiple

of |m(x) − C| 32 so that Theorem 1.5 applies.In particular, the case m(x) = 1, i.e., the bistable equation in the Introduction, falls within the range of our

results. The traveling waves for this model belong to the attractor and are actual connections. Note that muchstronger attracting properties are obtained here than in [21, 22].

In the same spirit, with the results in this paper we can consider nonlinearities of the form,

f(x, u) = m(x)u − n(x)u3 + g(x)

with m and n ≥ 0.

Then (1.13) is satisfied with C equal to a negative constant and D(x) is |g(x)| plus a multiple of(m(x)−C)

32+

n1/2(x).

Thus, Theorem 1.5 applies providedD ∈ LpU

(R

N).

Example 3.4 In [6] a result analogous to Theorem 1.5 was given in standard Lp(R

N)

spaces. In fact if (1.13)is satisfied with

C ∈ LpU

(R

N)

and D ∈ Lp(R

N),

where p > N2 , p ≥ 2, and the semigroup {e(∆+C(·)I)t} is asymptotically decaying, then Theorem 1.5 and the

results in [6] apply. Hence, the corresponding semigroup of global solutions is dissipative both in W 2α,pU

(R

N)↪→

L∞(R

N)

and in a Bessel potential space H2α,p(R

N)↪→ W 2α,p

U

(R

N), see [6]. Hence we denote here the

corresponding attractors AU and A respectively.We show now that in fact AU = A. Note, however, that A attracts in different norms solutions depending

whether the initial data is in LpU

(R

N)

or in Lp(R

N). Also, note that the inclusion A ⊂ AU is immediate.

Referring to the proof of Theorem 1.5 we observe that the absolute value of any solution to (2.22) will staybelow U , where

U = e(∆+C(·)I)t|u0| +∫ t

0

e(∆+C(·)I)(t−s)Dds, t > 0.

We remark that e(∆+C(·)I)t|u0| tends to zero in L∞(R

N)

as t → ∞ and, since 0 ≤ D ∈ Lp(R

N), the term∫ t

0 e(∆+C(·)I)(t−s)Dds converges in H2−ε,p

(R

N)

to a certain 0 ≤ Φ ∈ H2,p(R

N)

(see (2.23), (2.24) and(2.25)).

For any u ∈ AU we thus have

|u(x)| ≤ Φ(x), x ∈ RN .

In particular, AU is bounded in Lp(R

N) ∩ L∞(

RN).

With this, from (1.7), we have for every u0 ∈ AU and t > 0

u(t, u0) = eAtu0 +∫ t

0

eA(t−s)f(·, u(s, u0)) ds,

where f(·, u(s, u0)) is uniformly bounded in L∞(R

N) ∩ Lp

(R

N). Hence, using that AU is invariant, we get

from this, taking t = 1 that AU is actually bounded in H2α,p(R

N). Now the invariance and attractivity of A

implies AU ⊂ A.



Systems of semilinear parabolic equations can also be successfully treated within the approach of the presentpaper provided that a nonlinear vector field f = (f1, . . . , fN ) satisfies appropriate quasi-monotonicity assump-tion (“Kamke condition”, see [40], or “Wazewski condition”, see [42]).

Example 3.5 Let y = (y1, . . . , yN), y = (y1, . . . , yN) ∈ RN . We say that

y ≤ y if yj ≤ yj for j = 1, . . . , N,

and

yi≤ y if yj ≤ yj for j = 1, . . . , N and yi = yi.

Let f : RN → R

N be a locally Lipschitz continuous vector field which is moreover quasi-monotone increasing;that is for each i = 1, . . . , N,

yi≤ y implies that fi(y) ≤ fi(y).

Suppose that the system of ordinary differential equations

U = f(U), t > 0, U(0) = U0 ∈ RN , (3.6)

has an attractor in RN and consider the parabolic Cauchy problem{

ut = ∆u + f(u), t > 0, x ∈ RN ,

u(0) = u0 ∈ [W 1,pU

(R

N)]N

, p > N.(3.7)

If u(t,u0) is a solution to (3.7) and u0(x) belongs for each x ∈ RN to an ordered interval [ − M ,M ] ⊂

RN , then by comparison arguments (see e.g. [11, Lemma 3]) the solution u(t,u0) will be estimated above

and below by U(t,±M) as long as u(t,u0) exists. This ensures that the solutions to (3.7) are bounded in[L∞(

RN)]N

uniformly for t ≥ 0 and[L∞(

RN)]N

-estimate is asymptotically independent of initial condi-

tion u0 varying in bounded subsets of[W 1,p

U

(R

N)]N

. With such an estimate the proof of the existence of a

bounded dissipative semigroup {T (t)} corresponding to (3.7) on[W 1,p

U

(R

N)]N

and the existence of a bounded([W 1,p

U

(R

N)]N − [W 1,p

ρ

(R

N)]N)

attractor for {T (t)}, where ρ is any translation of ρ0(x) =(1 + |x|2)−ν

,

goes the same way as in the case of the scalar equation.Hence Corollary 1.4 can be formulated in the case of systems as well. Namely, for a locally Lipschitz con-

tinuous and quasi-monotone increasing vector field f there exists a bounded([W 1,p

U

(R

N)]N − [Lp

ρ0

(R

N)]N)

attractor for (3.7), iff there exists an attractor for (3.6) in RN .

A Appendix

In this section we recall some results reported in [7] that have been used before in this paper and prove an em-bedding property that has been used in the proof of Lemma 2.7. Moreover we give an additional characterizationof the spacesW s,p

U

(R

N). In particular we show that for non-integer orders W s,p

U

(R

N)

and W s,pU

(R

N)

coincide.As mentioned in the introduction Lp

U

(R

N)

is the Banach space of measurable functions φ : RN → R satisfy-

ing

‖φ‖LpU (RN ) := sup

y∈RN

‖φ‖Lp(B(y,1)) <∞

for 1 ≤ p ≤ ∞. The closed subspace of LpU

(R

N)

consisting of functions which are translation continuous withrespect to ‖ · ‖Lp

U (RN ), i.e.,

‖τyφ− φ‖LpU (RN ) −→ 0 as |y| −→ 0


Math. Nachr. 280, No. 15 (2007) 1659

where{τy , y ∈ R

N}

is the group of translations in RN , is denoted by Lp

U

(R

N).

Similarly, the Sobolev space, of integer order, constructed above LpU

(R

N), is denoted by W k,p

U

(R

N)

and it is

the Banach space consisting of elements φ ∈ W k,ploc

(R

N)

satisfying, for |σ| ≤ k, Dσφ ∈ LpU

(R

N), with norm

‖φ‖W k,pU (RN ) =

∑|σ|≤k ‖Dσφ‖Lp

U (RN ), or equivalently

‖φ‖W k,p

U

(RN) := sup

y∈RN

‖φ‖W k,p(B(y,1)) <∞.

Also, W k,pU

(R

N)

can be defined as the space of all φ ∈W k,ploc

(R

N)

satisfying, for |σ| ≤ k, Dσφ ∈ LpU

(R

N), or

equivalently, as the closed subspace of all elements in W k,pU

(R

N), which are translation continuous with respect

to ‖ · ‖W k,pU (RN ).

On the other hand, denoting by ρ any translation of the weight ρ0(x) =(1+ |x|2)−ν

, for ν > N2 , the weighted

space Lpρ

(R

N)

is defined as the Banach space of measurable functions such that

‖φ‖pLp

ρ(RN )=∫

RN

|φ(x)|pρ(x) dx <∞.

In fact, as shown in [7, Section 4], a very wide class of weights could be considered in the discussion rather thanρ0. However we stick to this example because it has been traditionally used in the literature.

Analogously the Sobolev space, of integer order, constructed on Lpρ

(R

N), is denoted W k,p

ρ

(R

N)

and it is

the Banach space consisting of elements φ ∈ W k,ploc

(R

N)

satisfying, for |σ| ≤ k, Dσφ ∈ Lpρ

(R

N), with norm

‖φ‖W k,pρ (RN ) =

∑|σ|≤k ‖Dσφ‖Lp

ρ(RN ).In summary, we have the following diagram, where all arrows indicate continuous inclusions

W k,pU

(R

N)

↪→ W k,pU

(R

N)

↪→ W k,pρ

(R

N)

↑ ↖ ↑ ↑W k+1,p

U

(R

N)

↪→ W k+1,pU

(R

N)

↪→ W k+1,pρ

(R

N)

where thanks to [7, Remark 4.5] we have W k+1,pU

(R

N)↪→ W k,p

U

(R

N).

Concerning the norms, it was also shown in [7] and [35] that, if ρ is any translation of the weight ρ0(x) =(1 + |x|2)−ν , ν > N

2 , the W k,pρ

(R

N)

space is isomorphic to a Bessel potential space, see [43], as

W k,pρ

(R

N)

= ρ1pHk

p

(R

N),

which gives an equivalent norm

‖φ‖W k,pρ (RN ) =

∥∥ρ 1pφ∥∥

Hkp (RN )

. (A.1)

Also an equivalent norm in W k,pU

(R

N)

is given by

‖φ‖W k,pU (RN ) := sup

y∈RN

‖φ‖W k,pρy (RN ) = sup

y∈RN

∥∥∥ρ 1py φ∥∥∥

Hkp (RN )

(A.2)

where ρy(x) =(1 + |x− y|2)−ν

, ν > N2 , see [7, Proposition 4.7].

For non-integer orders s ∈ (k, k + 1), the spaces W s,pU

(R

N), W s,p

U

(R

N), and W s,p

ρ

(R

N)

are defined as thecomplex interpolation spaces of the spaces with orders k and k + 1; hence

W s,pU

(R

N)↪→ W s,p

U

(R

N)↪→W s,p

ρ

(R

N).

In particular, from properties of interpolation, translations are continuous in W s,pU

(R

N)

with respect to the normof W s,p

U

(R

N).

For these spaces it is also shown in [7, 35] that

W s,pρ

(R

N)

= ρ1pHs

p

(R

N),



with norm

‖φ‖W s,pρ (RN ) =

∥∥ρ 1pφ∥∥

Hsp(RN )

.

On the other hand C∞bd

(R

N)

is dense in W s,pU

(R

N)

and an equivalent norm in this space is given by

‖φ‖W s,pU (RN ) := sup

y∈RN

‖φ‖W s,pρy (RN ) = sup

y∈RN

∥∥∥(ρy)1pφ∥∥∥

Hsp(RN )

(A.3)

see [7, Remark 5.7], while for φ ∈W s,pU

(R

N)

supy∈RN

‖φ‖W s,pρy (RN ) = sup

y∈RN

∥∥∥(ρy)1pφ∥∥∥

Hsp(RN )

≤ C‖φ‖W s,pU (RN ). (A.4)

Now we show the following characterization of the space W s,pU

(R

N).

Lemma A.1 For p ≥ 2 and s ≥ 0 the space W s,pU

(R

N)

coincides with

V0,s =

{φ ∈ ρ

1p

0 Hsp

(R

N)

: supy∈RN

∥∥∥ρ 1py φ∥∥∥

Hsp(RN )

<∞, lim|z|→0

supy∈RN

∥∥∥ρ 1py (τzφ− φ)

∥∥∥Hs

p(RN )= 0

}. (A.5)

P r o o f. Define the norm

‖φ‖V0,s = supy∈RN

∥∥∥ρ 1py φ∥∥∥

Hsp

(RN) (A.6)

which, by (A.2) and (A.3) is equivalent to the norm of W s,pU

(R

N)

and W s,pU

(R

N)

if s ∈ N, respectively. Inparticular the case s ∈ N follows from this property. Also, since C∞

bd

(R

N)

is dense in W s,pU

(R

N), for non-

integer s it will be then enough to show that C∞bd

(R

N)

is dense in V0,s.For this, following [17, § 4.2.1, 4.2.2] we define, for fixed s ≥ 0,

F(φ) =

⎛⎝ ∞∑j=0

22js∣∣F−1 (ψjFφ)

∣∣2⎞⎠12

, (A.7)

where F is a Fourier transform and {ψj} is the dyadic resolution of unity.Then, using mollifiers, we observe that for φ ∈ V0,s, Jε ∗ φ ∈ C∞

bd

(R

N)

and

1c

∥∥∥ρ 1py (Jε ∗ φ− φ)

∥∥∥Hs

p

(RN)

≤∥∥∥ρ 1

py F (Jε ∗ φ− φ)

∥∥∥Lp(RN )

=

∥∥∥∥∥∥∥ρ1py

⎛⎝ ∞∑j=0

22js∣∣Jε ∗ F−1(ψjFφ) − F−1(ψjFφ)

∣∣2⎞⎠12

∥∥∥∥∥∥∥Lp(RN )

=

⎛⎜⎝∫RN

⎧⎨⎩∞∑

j=0

22js

∣∣∣∣∫RN

Jε(z)[F−1(ψjFφ)(x − z) − F−1(ψjFφ)(x)

]dz

∣∣∣∣2⎫⎬⎭

p2

ρy(x) dx

⎞⎟⎠1p

≤

⎛⎜⎝∫RN

⎧⎨⎩∞∑

j=0

22js

∫RN

Jε(z) dz∫

RN

Jε(z)|F−1(ψjFφ)(x− z) − F−1(ψjFφ)(x)|2 dz⎫⎬⎭

p2

ρy(x) dx

⎞⎟⎠1p

≤(∫

RN

{∫RN

Jε(z)F2(φ(x − z) − φ(x))ρ2py (x) dz

} p2

dx

) 1p

.


Math. Nachr. 280, No. 15 (2007) 1661

Applying then the generalized Minkowski inequality we obtain the estimate

1c

∥∥∥ρ 1py (Jε ∗ φ− φ)

∥∥∥Hs

p(RN )≤(∫

RN

{∫RN

Jp2ε (z)Fp(φ(x − z) − φ(x))ρy(x) dx

} 2p

dz

) 12

=(∫

RN

Jε(z)∥∥∥ρ 1

py F(φ(· − z) − φ)

∥∥∥2Lp(RN )

dz

) 12

≤ sup|z|<ε

∥∥∥ρ 1py F(φ(· − z) − φ)

∥∥∥Lp(RN )

(∫RN

Jε(z) dz) 1

2

≤ const . sup|z|<ε

∥∥∥ρ 1py (φ(· − z) − φ)

∥∥∥Hs

p(RN ).

Hence

‖Jε ∗ φ− φ‖V0,s ≤ C sup|z|<ε

‖(φ(· − z) − φ)‖V0,s , (A.8)

which shows that Jε ∗ φ converges to φ as ε→ 0 for any φ ∈ V0,s.

Note that inequality (A.8) generalizes for non-integer s the one obtained in [7, Lemma 4.9] for W k,pU

(R

N).

As a consequence we get the following description of W s,pU

(R

N)

spaces, which has been used in the proof ofLemma 2.7.

Corollary A.2 For p ≥ 2 and s ≥ 0 the space W s,pU

(R

N)

coincides with the closed subspace of W s,pU

(R

N)

given by

Vs = {φ ∈W s,pU

(R

N),(τyφ− φ

) −→ 0 in W s,pU

(R

N)

as y −→ 0}.In particular,

W k,pU

(R

N)↪→ W s,p

U

(R

N)

for 0 ≤ s < k, k ∈ N, p ≥ 2, (A.9)

and for non-integer s and p ≥ 2

W s,pU

(R

N)

= W s,pU

(R

N).

P r o o f. From (A.4) and the fact that in W s,pU

(R

N)

translations are continuous in the norm of W s,pU

(R

N),

note that W s,pU

(R

N) ⊂ Vs ⊂ V0,s. Therefore the first assertion holds as a consequence of Lemma A.1.

Now, to prove (A.9), we show that W k+1,pU

(R

N)↪→ W s,p

U

(R

N)

for s ∈ (k, k + 1). From the char-acterization above, it is enough to prove the continuity of translations with respect to W s,p

U

(R

N)-norm, for

w ∈ W k+1,pU

(R

N). By [7, Remark 4.5] we have W k+1,p

U

(R

N)↪→ W k,p

U

(R

N), whereas W s,p

U

(R

N)

=[W k,p

U

(R

N),W k+1,p

U

(R

N)]

θwhenever θ ∈ (0, 1) and s = k+ θ ∈ (k, k+ 1). Then, for w ∈W k+1,p

U

(R

N), by

the interpolation inequality we obtain

‖τyw − w‖W s,pU (RN ) ≤ c ‖τyw − w‖θ

W k+1,pU (RN )

‖τyw − w‖1−θ

W k,pU (RN )

≤ C ‖τyw − w‖1−θ

W k,pU (RN )

−→ 0 as y −→ 0.

Finally, we prove W s,pU

(R

N)

= W s,pU

(R

N)

for non-integer s. Observe that the inclusion W s,pU

(R

N)↪→

W s,pU

(R

N)

is obvious. Next if k ∈ N and k < s < k + 1 then by properties of the complex interpolation

W k+1,pU

(R

N)↪→ W s,p

U

(R

N)

is dense in W s,pU

(R

N), see [43, Theorem 1.9.3 (c)] and we get the result.

Acknowledgements The authors were partially suported by Project BFM2003–03810, DGES Spain and by KBN Grant 2P03A 035 18, Poland.



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