Global Attractors in H1 ( ...N ) for Nonclassical Diffusion Equations

Abstract

We study the existence of global attractors for nonclassical diffusion equations in [superscript] H1[/superscript] ( [superscript] ...N[/superscript] ) . The nonlinearity satisfies the arbitrary order polynomial growth conditions.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Chuanxi Qian

College of Mathematics and Information Science, Northwest Normal University, Gansu, Lanzhou 730070, China

Received 14 May 2012; Accepted 22 October 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations: [figure omitted; refer to PDF] with the initial data [figure omitted; refer to PDF] where g (x ) ∈ ^{L 2} ( ^{... N} ) , and the nonlinearity f (x ,u ) = _{f 1} (u ) +a (x ) _{f 2} (u ) satisfies

: ( _{F 1} ) _{α 1} |u ^{| p} - _{β 1} |u ^{| 2} ...4; _{f 1} (u ) (u ) ...4; _{γ 1} |u ^{| p} + _{δ 1} |u ^{| 2} , _{f 1} (u )u ...5;0 , p ...5;2 , and ^{f 1 [variant prime]} (u ) ...5; -c ,

: ( _{F 2} ) _{α 2} |u ^{| p} - _{β 2} ...4; _{f 2} (u ) (u ) ...4; _{γ 2} |u ^{| p} + _{δ 2} , p ...5;2 , and ^{f 2 [variant prime]} ( u ) ...5; -c ,

: and

: ( A ) a ∈ ^{L 1} ( ^{... N} ) ∩ ^{L ∞} ( ^{... N} ) , a (x ) >0 ,

where _{α i} , _{β i} , _{γ i} , _{δ i} , i =1,2 , and c are all positive constants. Moreover, without loss of generality, we also assume _{f 1} (0 ) = _{f 2} (0 ) =0 .

In 1980, Aifantis in [ 1- 3] pointed out that the classical reaction-diffusion equation [figure omitted; refer to PDF] does not contain each aspect of the reaction-diffusion problem, and it neglects viscidity, elasticity, and pressure of medium in the process of solid diffusion and so forth. Furthermore, Aifantis found out that the energy constitutional equation revealing the diffusion process is different along with the different property of the diffusion solid. For example, the energy constitutional equation is different, when conductive medium has pressure and viscoelasticity or not. He constructed the mathematical model by some concrete examples, which contains viscidity, elasticity, and pressure of medium, that is the following nonclassical diffusion equation: [figure omitted; refer to PDF] This equation is a special form of the nonclassical diffusion equation used in fluid mechanics, solid mechanics, and heat conduction theory (see [ 1- 4]). Recently, Aifantis presented a new model about this problem and scrutinized the concrete process of constructing model; the reader can refer to [ 5] for details.

The longtime behavior of ( 1.1) acting on a bounded domain Ω has been extensively studied by several authors in [ 6- 13] and references therein. In [ 12] the existence of a global attractor for the autonomous case has been shown provided that the nonlinearity is critical and g (x ) ∈ ^{H -1} ( Ω ) . Furthermore, for the non-autonomous, the existence of a uniform attractor and exponential attractors has been scrutinized when the time-dependent forcing term g (x ,t ) only satisfies the translation bounded domain instead of translation compact, namely, g (x ,t ) ∈ ^{L b 2} ( ... , ^{L 2} ( Ω ) ) . A similar problem was discussed in [ 13] by virtue of the standard method based on the so-called squeezing property. To our best knowledge, the dynamics of ( 1.1) acting on an unbounded domain ^{... N} has not been considered by predecessors.

As we know, if we want to prove the existence of global attractors, the key point is to obtain the compactness of the semigroup in some sense. For bounded domains, the compactness is obtained by a priori estimates and compactness of Sobolev embeddings. This method does not apply to unbounded domains since the embeddings are no longer compact. To overcome the difficulty of the noncompact embedding, in [ 14], using the idea of Ball [ 15], the author proved that the solutions are uniformly small for large space and time variables and then showed that the weak asymptotic compactness is equivalent to the strong asymptotic compactness in certain circumstances. In [ 16], the authors provided new a priori estimates for the existence of global attractors in unbounded domains and then applied this approach to a nonlinear reaction-diffusion equation with a nonlinearity having a polynomial growth for arbitrary order p -1 (p ...5;2 ) and with distribution derivatives in homogeneous term. More recently, in [ 17] the authors achieved the existence of global attractors for reaction-diffusion equations in ^{L 2} ( ^{... n} ) , by using the methods presented in [ 18]. Our purpose in this paper is to study the existence of global attractors of ( 1.1) on the unbounded domains ^{... n} , and we borrow the idea of [ 17, 18]. Our main result is Theorem 4.6.

This paper is organized as follows. In Section 2, we recall some basic definitions and related theorems that will be used later. In Section 3, we prove the existence of weak solution for nonclassical diffusion equations in ^{H 1} ( ^{... N} ) . The main result is stated and proved in Section 4.

2. Preliminaries

In this section, we iterate some notations and abstract results.

Definition 2.1 (see [ 18]).

Let M be a metric space, and let A be bounded subsets of M . The Kuratowski measure of noncompactness γ (A ) of A defined by [figure omitted; refer to PDF]

Definition 2.2 (see [ 18]).

Let X be a Banach space, and let {S (t ) _{} t ...5;0} be a family of operators on X . We say that {S (t ) _{} t ...5;0} is a continuous semigroup ( _{C 0} semigroup) (or norm-to-weak continuous semigroup) on X , if {S (t ) _{} t ...5;0} satisfies

(i) S (0 ) =Id (the identity),

(ii) S (t )S (s ) =S (t +s ) , for all t , s ...5;0 ,

(iii): S ( _{t n} ) _{x n} [arrow right]S (t )x , if _{t n} [arrow right]t , _{x n} [arrow right]x in X (or ( iii ) S ( _{t n} ) _{x n} ...S (t )x , if _{t n} [arrow right]t , _{x n} [arrow right]x in X ).

Definition 2.3 (see [ 18]).

A _{C 0} semigroup (or norm-to-weak continuous semigroup) {S (t ) _{} t ...5;0} in a complete metric space M is called ω -limit compact if for every bounded subset B of M and for every [straight epsilon] >0 , there is a t (B ) >0 , such that [figure omitted; refer to PDF]

Condition C (see [ 18]).

For any bounded set B of a Banach space X , there exists a t (B ) >0 and a finite dimensional subspace _{X 1} of X such that { || _{P m} S (t )B || } is bounded and [figure omitted; refer to PDF] where _{P m} :X [arrow right] _{X 1} is a bounded projector.

Lemma 2.4 (see [ 18]).

Let X be a Banach space, and let {S (t ) _{} t ...5;0} be a _{C 0} semigroup (or norm-to-weak continuous semigroup) in X .

(1) If Condition C holds, the {S (t ) _{} t ...5;0} is ω -limit compact.

(2) Let X be a uniformly convex Banach space. Then {S (t ) _{} t ...5;0} is ω -limit compact if and only if Condition C holds.

Lemma 2.5 (see [ 18]).

Let X be a Banach space, and let {S (t ) _{} t ...5;0} be a _{C 0} semigroup (or norm-to-weak continuous semigroup) in X .

(1) If Condition C holds, the {S (t ) _{} t ...5;0} is ω -limit compact;

(2) Let X be a uniformly convex Banach space. Then {S (t ) _{} t ...5;0} is ω -limit compact if and only if Condition C holds.

Theorem 2.6 (see [ 18]).

Let X be a Banach space. Then the _{C 0} semigroup (or norm-to-weak continuous semigroup) {S (t ) _{} t ...5;0} has a global attractor in X if and only if

(1) there is a bounded absorbing set B ⊂X .

(2) {S (t ) _{} t ...5;0} is ω -limit compact.

Lemma 2.7 (see [ 19]).

Let Φ be an absolutely continuous positive function on ^{... +} , which satisfies for some [straight epsilon] >0 the differential inequality [figure omitted; refer to PDF] for almost every t ∈ ^{... +} , where g and h are functions on ^{... +} such that [figure omitted; refer to PDF] for some _{m 1} ...5;0 and μ ∈ [0,1 ) , and [figure omitted; refer to PDF] for some _{m 2} ...5;0 . Then [figure omitted; refer to PDF] for some β = β ( _{m 1} , μ ) ...5;1 and [figure omitted; refer to PDF]

Lemma 2.8 (see [ 20]).

Let X ⊂ ⊂H ⊂Y be Banach spaces, with X reflexive. Suppose that _{u n} is a sequence that is uniformly bounded in ^{L 2} (0 ,T ;X ) , and d _{u n} /dt is uniformly bounded in ^{L p} (0 ,T ;Y ) , for some p >1 . Then there is a subsequence that converges strongly in ^{L 2} (0 ,T ;H ) .

3. Unique Weak Solution

Theorem 3.1.

Assume ( _{F 1} ) , ( _{F 2} ) , and (A ) are satisfied. Then for any T >0 and _{u 0} ∈ ^{H 1} ( ^{R N} ) , there is a unique solution u of ( 1.1)-( 1.2) such that [figure omitted; refer to PDF] Moreover, the solution continuously depends on the initial data.

Proof.

We decompose our proof into three steps for clarity.

Step 1 . For any n ∈N , we consider the existence of the weak solution for the following problem in B (0 ,n ) [triangle, =] _{B n} ⊂ ^{R N} : [figure omitted; refer to PDF] Choose a smooth function _{χ n} (x ) with [figure omitted; refer to PDF]

Since _{B n} is a bounded domain, so the existence and uniqueness of solutions can be obtained by the standard Faedo-Galerkin methods; see [ 6, 8, 11, 16]; we have the unique weak solution [figure omitted; refer to PDF]

Step 2. According to Step 1, we denote (d /dt ) _{u n} = _{u nt} ; then _{u n} satisfies [figure omitted; refer to PDF] For the mathematical setting of the problem, we denote _{|| · || ^{L 2} ( B n )} [triangle, =] _{|| · || B n} , _{|| · || ^{L 1} ( ^{R N} )} [triangle, =] _{|| · || 1} , _{|| · || ^{L 2} ( ^{R N} )} [triangle, =] || · || , _{|| · || ^{L ∞} ( ^{R N} )} [triangle, =] _{|| · || ∞} .

Multiplying ( 3.5) by _{u n} in _{B n} , using _{f 1} ( u ) u ...5;0 , ( _{F 2} ) and (A ) , we have [figure omitted; refer to PDF] By the Poincaré inequality, for some ν >0 , we conclude that [figure omitted; refer to PDF] Hence, it follows that [figure omitted; refer to PDF] We get the following estimate: [figure omitted; refer to PDF] Similar to ( 3.9), using ( _{F 1} ) , ( _{F 2} ) , and (A ) , we get [figure omitted; refer to PDF] where C is independent of n .

( _{F 1} ) and ( _{F 2} ) yield [figure omitted; refer to PDF] Choose q such that (1 /p ) + (1 /q ) =1 ; then (p -1 )q =p . Noting that p ...5;2 , then 1 <q ...4;2 , and we have the embedding ^{L p} ( _{B n} ) ... ^{L q} ( _{B n} ) . According to ( 3.12) and ( 3.13), we get [figure omitted; refer to PDF] where C is independent of n .

Thanks to ( 3.14), _{f 1} ( _{u n} ) is bounded in ^{L p} (0 ,T ; ^{L q} ( _{B n} ) ) , and a _{f 2} ( _{u n} ) is bounded in ^{L p} (0 ,T ; ^{L q} ( _{B n} ) ) .

For ∀v ∈ ^{L 2} (0 ,T ; ^{H 0 1} ( _{B n} ) ) , [figure omitted; refer to PDF] where C is independent of n . We can obtain that - Δ _{u n} is bounded in ^{L 2} (0 ,T ; ^{H -1} ( _{B n} ) ) .

Since g (x ) ∈ ^{L 2} ( ^{... N} ) , [figure omitted; refer to PDF] Therefore, there exists s >0 , such that ^{L 2} (0 ,T ; ^{H -1} ( _{B n} ) ) , ^{L 2} (0 ,T ; ^{H 0 1} ( _{B n} ) ) , ^{L q} (0 ,T ; ^{L q} ( _{B n} ) ) , and ^{L 2} (0 ,T ; ^{L 2} ( _{B n} ) ) are continuous embedding to ^{L q} (0 ,T ; ^{H -s} ( _{B n} ) ) .

According to ( 3.5) and ( 3.14)-( 3.16), we obtain [figure omitted; refer to PDF] So _{u n} has a subsequent (we also denote _{u n} ) weak* convergence to u in ^{L 2} (0 ,T ; ^{H -1} ( _{B n} ) ) and ^{L 2} (0 ,T ; ^{L 2} ( _{B n} ) ) ; _{u nt} - Δ _{u nt} has a subsequent (we also denote _{u nt} - Δ _{u nt} ) weak* convergence to _{u t} - Δ _{u t} . Let _{u n} =0 outside of _{B n} ; we can extend _{u n} to ^{... N} .

As introduced in [ 6, 20], ^{C c ∞} ( ^{... N} ) is dense in the dual space of ^{H -1} ( _{B n} ) , ^{L 2} ( _{B n} ) , ^{L q} ( _{B n} ) , and ^{H -s} ( _{B n} ) , so we can choose for all [varphi] ∈ ^{L 2} (0 ,T ; ^{C c ∞} ( ^{... N} ) ) ∩ ^{L q} (0 ,T ; ^{C c ∞} ( ^{... N} ) ) as a test function such that [figure omitted; refer to PDF]

Since for all [varphi] ∈ ^{C c ∞} ( ^{... N} ) , there exists bounded domain Ω ⊂ ^{... N} such that [varphi] =0 , x ∉ Ω . It follows that _{u n} is uniformly bounded in ^{L 2} (0 ,T ; ^{H 0 1} ( Ω ) ) , and _{u nt} - Δ _{u nt} ∈ ^{L q} (0 ,T ; ^{H -s} ( Ω ) ) . Since ^{H 0 1} ( Ω ) ⊂ ⊂ ^{L 2} ( Ω ) ⊂ ^{H -s} ( Ω ) , according to Lemma 2.8, there is a subsequence _{u n} (we also denote _{u n} ) that converges strongly to u in ^{L 2} (0 ,T ; ^{L 2} ( Ω ) ) .

Using the continuity of _{f 1} and _{f 2} , we have [figure omitted; refer to PDF]

In line with ( 3.18) and ( 3.19), and let n [arrow right] ∞ , we geting for all [varphi] ∈ ^{L 2} (0 ,T ; ^{C c ∞} ( ^{... N} ) ) ∩ ^{L q} (0 ,T ; ^{C c ∞} ( ^{... N} ) ) : [figure omitted; refer to PDF] Thus, u is the weak solution of ( 3.2) and satisfies [figure omitted; refer to PDF]

Step 3 (uniqueness and continuous dependence). Let _{u 0} , _{v 0} be in ^{H 1} ( ^{... N} ) , and setting w (t ) =u (t ) -v (t ) , we see that w (t ) satisfies [figure omitted; refer to PDF] Taking the inner product with w of ( 3.22), using ( _{F 1} ) , ( _{F 2} ) , and (A ) , we obtain [figure omitted; refer to PDF] By the Gronwall Lemma, we get [figure omitted; refer to PDF] This is uniqueness and is continuous dependence on initial conditions.

Thanks to Theorem 3.1, and leting S (t ) _{u 0} =u (t ) , S (t ) : ^{H 1} ( ^{... N} ) [arrow right] ^{H 1} ( ^{... N} ) is a ^{C 0} semigroup.

4. Global Attractor in ^{... N}

Lemma 4.1.

Assume ( _{F 1} ) , ( _{F 2} ) , and (A ) are satisfied. There is a positive constant _{ρ 1} such that for any bounded subset B ⊂ ^{H 1} ( ^{... N} ) , there exists _{T 1} = _{T 1} (B ) such that [figure omitted; refer to PDF]

Proof.

Multiplying ( 1.1) by u in ^{... N} , using _{f 1} (u )u ...5;0 , ( _{F 2} ) and (A ) , we have [figure omitted; refer to PDF] By virtue of the Poincaré inequality, for some ν >0 , there holds [figure omitted; refer to PDF] Furthermore, [figure omitted; refer to PDF] By the Gronwall Lemma, we get [figure omitted; refer to PDF] We completed the proof.

According to Lemma 4.1, we know that [figure omitted; refer to PDF] is a compact absorbing set of a semigroup of operators _{{S (t ) } t ...5;0} generalized by ( 1.1)-( 1.2), (F1 ) , (F2 ) , and (A ) .

Lemma 4.2.

Assume ( _{F 1} ) , ( _{F 2} ) , and (A ) hold. Then for any _{u 0} ∈ ^{H 1} ( ^{... N} ) and [varepsilon] >0 , there are some T ( [varepsilon] ) and k ( [varepsilon] ) such that [figure omitted; refer to PDF] whenever k ...5;T ( [varepsilon] ) and t ...5;t ( [varepsilon] ) .

Proof.

Choose a smooth function θ (x ) with [figure omitted; refer to PDF] where 0 ...4; θ (s ) ...4;1 , 1 ...4;s ...4;2 , and there is a constant c such that | ^{θ [variant prime]} (s ) | ...4;c .

Multiplying ( 1.1) with ^{θ 2} ( |x ^{| 2} / ^{k 2} )u and integrating on ^{... N} , we obtain [figure omitted; refer to PDF] Next we deal with the right hand side of ( 4.9) one by one: [figure omitted; refer to PDF] According to the condition | ^{θ [variant prime]} (s ) | ...4;c and the bounded absorbing set in ^{H 1} ( ^{... N} ) for t ...5; _{t *} , it follows that [figure omitted; refer to PDF] where C is independent of k . For any 0 < [varepsilon] <1 given, let [figure omitted; refer to PDF]

Hence, combining ( 4.10) with ( 4.11), when k ...5; _{k 1} ( [varepsilon] ) , we conclude that [figure omitted; refer to PDF] Using _{f 1} (u )u ...5;0 and ( _{F 2} ) , it yields [figure omitted; refer to PDF] Since a ∈ ^{L 1} ( ^{... N} ) , there exist _{k 2} ( [varepsilon] ) > _{k 1} ( [varepsilon] ) , such that [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] From the assumption g (x ) ∈ ^{L 2} ( ^{... N} ) , provide k ...5;k ( [varepsilon] ) ...5; _{k 2} ( [varepsilon] ) , such that [figure omitted; refer to PDF] Thus combining ( 4.9), ( 4.13), ( 4.16), and ( 4.17), we finally obtain [figure omitted; refer to PDF] Furthermore, there holds [figure omitted; refer to PDF] According to Lemma 2.7, we obtain [figure omitted; refer to PDF] Thus, we get [figure omitted; refer to PDF] provided T ...5;T ( [varepsilon] ) and k ...5; k ~ ( [varepsilon] ) , we complete the proof.

Lemma 4.3.

Assume ( _{F 1} ) , ( _{F 2} ) , and (A ) hold. There is a positive constant _{ρ 2} such that for any bounded subset B ⊂ ^{H 2} ( ^{... N} ) , there exists _{T 2} = _{T 2} (B ) such that [figure omitted; refer to PDF]

Proof.

Multiplying ( 1.1) by - Δu in ^{... N} , we find [figure omitted; refer to PDF] Using ( _{F 1} ) , ( _{F 2} ) , and (A ) , we have the following estimates: [figure omitted; refer to PDF] Together with ( 4.6) and ( 4.19)-( 4.21), by the Poincaré inequality, for some μ >0 , this yields [figure omitted; refer to PDF] By the Gronwall Lemma, we get [figure omitted; refer to PDF] We complete the proof.

Remark 4.4.

There is a constant C >0 , such that for any bounded subset B ⊂B (0 , _{ρ 2} ) ⊂ ^{H 1} ( ^{... N} ) , when t > _{t *} , there holds [figure omitted; refer to PDF]

Lemma 4.5.

Assume ( _{F 1} ) , ( _{F 2} ) , and (A ) are satisfied. Then the semigroup {S (t ) _{} t ...5;0} associated with the initial value problems ( 1.1) and ( 1.2) is ω -limit compact.

Proof.

Denote _{B R} =B (0 ;R ) ∩ ^{... N} , and we split u (t ) as [figure omitted; refer to PDF] where θ (x ) is a smooth function: [figure omitted; refer to PDF] with 0 ...4; χ (x ) ...4;1 , and there is a positive constant c such that | ^{χ [variant prime]} (x ) | ...4;c . Then [figure omitted; refer to PDF] From Lemma 4.1, we know that _{u 1} (t ) ∈ ^{H 1} ( _{B R} ) as t ...5; _{T 1} .

For any [varepsilon] >0 given, we can choose R large enough; by Remark 4.4, we can assume [figure omitted; refer to PDF] So we conclude that [figure omitted; refer to PDF] For any bounded set B ⊂ ^{H 1} ( ^{... N} ) , {S (t )B _{} t ...5;0} = _{{ S ( t ) u 0 |" u 0 ∈B } t ...5;0} can be split as [figure omitted; refer to PDF] Then in line with the property of noncompact measure, it follows that [figure omitted; refer to PDF] On the other hand, [figure omitted; refer to PDF] From Lemma 4.3, we get [figure omitted; refer to PDF] Recall that [figure omitted; refer to PDF] On account of Remark 4.4, it yields [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF] That is, {S (t ) _{} t ...5;0} is ω -limit compact in ^{H 1} ( ^{... N} ) .

Theorem 4.6.

Assume ( _{F 1} ) , ( _{F 2} ) , and (A ) hold. Then the semigroup {S (t ) _{} t ...5;0} associated with the initial value problems ( 1.1) and ( 1.2) has a global attractor ...9C; in ^{H 1} ( ^{... N} ) .

Acknowledgments

The authors would like to thank the referee for careful reading of the paper and for his or her many vital comments and suggestions. This work was partly supported by the NSFC (11061030,11101334) and the NSF of Gansu Province (1107RJZA223), in part by the Fundamental Research Funds for the Gansu Universities.

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