Asymptotic Behavior for a Class of Nonclassical Parabolic Equations

This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations u t − εΔu t − ωΔu + f(u) = g(x) with critical nonlinearity, where ε ∈ [0, 1] and ω > 0 are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for g(x) ∈ H(Ω), which are independent of the parameter ε. Secondly, some uniformly (with respect to ε ∈ [0, 1]) asymptotic regularity about the solutions has been established for g(x) ∈ L(Ω), which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ε ∈ [0, 1]). Finally, as an application of this regularity result, a family {E ε } ε∈[0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (ε = 0), the upper semicontinuity, at ε = 0, of the global attractors has been proved.

For the fixed constant (>0), any  ∈ [0, 1], and the longtime behavior of the solutions of (  ) has been considered by some researchers; see [10,13].In [10] the author proved the existence of a class of attractors in  2 ∩  1 0 with initial data  0 ∈  2 ∩  1 0 and the upper semicontinuity of attractors in  1  0 under subcritical assumptions and () = 0 in the case of  ≤ 3.In [13] similar results have been shown when  ≥ 3 and () ∈  1 0 (Ω).In this paper, inspired by the ideas in [17,18] and motivated by the dynamical results in [19][20][21][22], we study the uniform (with respect to the parameter  ∈ [0,1]) qualitative analysis (a priori estimates) for the solutions of the nonclassical parabolic equations (  ) and then give some information about the relation between the solutions of ( 0 ) and those of (  ).Our main difficulty comes from the critical nonlinearity and the uniformness with respect to  ∈ [0, 1].
This paper is organized as follows.In Section 2, we introduce basic notations and state our main results.In Section 3, we recall some abstract results that we will use later.In Section 4, we present several dissipative estimates about the solution of (  ) when () ∈  −1 (Ω), which hold uniformly with respect to  ∈ [0, 1].The main results are proved for () ∈  2 (Ω) in Section 5.Moreover, in Section 6, as an application, we construct a finite dimensional exponential attractor and prove the upper semicontinuity of the global attractor obtained in Section 5.

Main Results
Before presenting our main results, we first state the basic mathematical assumptions for considering the long-time behaviors of the nonclassical parabolic equations and then introduce some notations that we will use throughout this paper.
From the work in [18,19], we know that a very large damping has the effect of freezing the system, if the damping acts only on the velocity   , and this prevents the squeezing of the component .Therefore, the most dissipative situation occurs in between, that is, for a certain damping  * , which depends on the other coefficient of the equation.Therefore, in our frame, we choose  > 1 such that 1/ <  as  ∈ [0, 1] in order to obtain the uniformly (with respect to  ∈ [0, 1]) asymptotic regularity about the solutions of (  ).
(iii)  = −Δ with domain () =  2 ∩  1 0 , and consider the family of Hilbert space ( /2 ),  ∈ R with the standard inner products and norms, respectively, In particular, ⟨⋅, ⋅⟩ and ‖ ⋅ ‖ mean the  2 (Ω) inner product and norm, respectively. (iv) In particular, we denote and define H   as Then The global well-posedness of solutions and its asymptotic behavior for (  0 ) have been studied extensively under assumptions (1)-( 2) by many authors in [9][10][11][12][13][14] and references therein in fact note that The main results of this paper are the following asymptotic regularity.
Theorem 1.Under assumptions (1), (2), and  > 1, there exist a positive constant ], a bounded (in H 1 ) subset B ⊂ H 1 , and a continuous increasing function where B, ], and (⋅) are all independent of , and {  ()} ≥0 is the semigroup generated by This result says that asymptotically, for each (  ), the solutions are exponentially approaching a more regular fixed subset B uniformly (with respect to ∈ [0, 1]) for  > 1.Moreover, it implies the following results.
(1) For each  ∈ [0, 1], {  ()} ≥0 has a global attractor A  in H, and (2) Based on Theorem 1, applying the abstract result devised in [23,24], for each  ∈ [0, 1] we can prove the existence of a finite dimensional exponential attractor For the proof of Theorem 1, the main difficulty comes from the critical nonlinearity and the uniformness with respect to  ∈ [0, 1].
Hereafter, we will also use the following notation: denote by J the space of continuous increasing functions  : [0,∞) → [0,∞) and by D the space of continuous decreasing functions  : [0, ∞) → [0, ∞) such that (∞) < 1.Moreover, ,   , and   are the generic constants, and (⋅),   (⋅) ∈ J are generic functions, which are all independent of ; otherwise we will point out clearly.

Preliminaries
In this section, we recall some results used in the main part of the paper.
The first result comes from [17], which will be used to prove the asymptotic regularity for the case () ∈  2 (Ω).
Lemma 2 (see [17]).Let  and  be two Banach spaces and {()} ≥0 a  0 -semigroup on  with a bounded absorbing set  ⊂ .For every  ∈ , assume that there exist two solution operators   () on  and   () on  satisfying the following properties.
Next, we recall a criterion for the upper semicontinuity of attractors.
Lemma 3 (see [25,26]).Let {  ()} ≥0 be a family of semigroups defined on the Banach space , and for each  ∈ Λ, let {  ()} ≥0 have a global attractor A  .Assume further that  0 is a nonisolated point of Λ and that there exist  > 0,  0 > 0, and a compact set  ⊂  such that Then the global attractors A  are upper semicontinuous on Λ at  =  0 ; that is, Lemma 4 (see [27]).Let Φ be an absolutely continuous positive function on R + , which satisfies for some  > 0 the differential inequality for almost every  ∈ R + , where ℎ 1 and ℎ 2 are functions on R + such that for some  1 ≥ 0 and  ∈ [0, 1], and for some  2 ≥ 0. Then for some  1 =  1 ( 1 , ) ≥ 1 and For the proof, we refer the reader to [27,Lemma 2.2].
A standard Gronwall-type lemma will also be needed.
Lemma 5. Let Ψ be an absolutely continuous positive function on R + , which satisfies for some Ψ > 0 the differential inequality for some , ,  > 0 and some  ∈ J.Then,
The main purpose of this section is to deduce some dissipative estimates about the semigroups {  ()} ≥0 ( ∈ [0, 1]) associated with (  ) in H. Here, using the method in [19,20,22] for a strongly damped wave equation and a semilinear second order evolution equation, we will show that the radius of the absorbing set of {  ()} ≥0 associated with (  ) in H can be chosen to be independent of  ∈ [0, 1].Lemma 6.There exists a positive constant , which depends only on , ‖‖  −1 , and coefficients of (1)-( 2 where  1 =  1 (‖‖ H 0  ) depends on ‖‖ H 0  but not on .
Remark 7. Observing that above process of proof, we can also deduce that, for any  ∈ [0, 1] and any  ⊂ where (⋅) ∈ J is independent of  and .Moreover, if  is bounded in H, then we can obtain for some constant  ,‖‖ H which depends on , ‖‖ H . Indeed, from the fact that there is a constant can be obtained just by repeating the proof of Lemma 6 and taking  0 = 0 in (35) since  is bounded in H.
On the other hand, from the proof of Claim 3 as follows, we can get further estimates about Lemma 8.There exists a positive constant  3 such that for any  ∈ [0, 1] and any bounded (in where () =   () 0 ,  0 ∈ ,  1 is the time given in Claim 1, and  3 only depends on  but is independent of  and .
Hereafter, we denote the uniformly (with respect to  ∈ [0, 1]) bounded absorbing set obtained in Lemma 6 as  0 , that is, and denote the time by Λ 0 such that Lemmas 6 and 8 hold for  0 ; that is, holds for any  ∈ [0, 1] and all  ≥ Λ 0 .Moreover, similar to Remark 7, noting now that  0 is bounded in H, we have
Lemma 11.Let {()} ≥0 be a continuous semigroup on the Banach space , satisfying Then Its proof is obvious and we omit it here.The next estimate is about the solution of (60).
Proof.Multiplying (60) by   () and integrating over Ω, Then the proof is completely similar to that in [12,Lemma 3.4], so, we omit it.
Based on Lemmas 10 and 12, following the idea in Zelik [21], we can now decompose () as follows.
Step 2. We claim that there exists a constant   > 0 which depends only on   such that ) .
Then using the Gronwall inequality and integrating over [0, ] (from Lemma 12), we obtain Taking  (in Lemma 13) small enough such that  < /2 4 , we have Thus, Substituting above (100) and ( 102) into (99), we get that for all  ≥ 0       /2 2 :=   . (103) Step 3. Based on Step 1 and Step 2, applying the attraction transitivity lemma given in [28, Theorem 5.1] and noticing the Holder continuity Lemma 9, we can prove our lemma by performing a standard bootstrap argument, whose proof is now simple since Step 1 makes the nonlinear term become subcritical to some extent.

Proof of Theorem 1.
Lemma 14 has shown some asymptotic regularities; however, the radius of ‖  ‖ H 1 depends on  and the distances only under the H 0  -norm.To prove Theorem 1, we first give two lemmas as preliminary.
Lemma 15.There exsits a constant  1 > 0 such that for any bounded (in Proof.Multiplying (  ) by −Δ, we find hence, we obtain where  2 is a small, positive constant.Similarly, with using Lemma 4 we finally complete the proof.

Proof of Theorem 1. Set
where the constant  2 comes from Lemma 16.
From Lemmas 16 and 14, we know that there is a  0 such that   ()  ⊂ B (recall that   is given in (78)) for all  ≥  0 and any  ∈ [0, 1].
On the other hand, note that ∃ 1 ,  2 > 0 such that Then, from Lemma 9, there exists Hence, noting that  0 ,  1 , and  are all fixed, we can complete the proof by taking ] = ]/2 and applying Lemma 11.

Applications of Theorem 1
As for the applications of Theorem 1, in this subsection, we consider the existence of finite dimensional exponential attractors and the upper semicontinuity of global attractors for problem (  ) under assumptions (1), (2), and  > 1.

A Priori Estimates.
For the subset B defined in (113), and from Lemmas 6 and 8 we know that there is a  B such that where () =   () 0 .Now, for each  ∈ [0, 1], define B as follows: where Next, for clarity, we decompose the remainder proof into two steps.
Proof.For each  ∈ [0,1], we know that B is invariant and compact in H 0  .Hence, applying the abstract results established in [23,24], from Lemmas 17 and 18 we can first construct an exponential attractor on B with respect to the H 0  -norm.Then, we can complete the proof by using the attraction transitivity lemma given in [28, Theorem 5.1] from Lemma 14 and the Hölder continuity (47).(141) Proof.Since the global attractor A  is strictly invariant, that is,   ()A  = A  for all  ≥ 0, it is obvious to see that A  ⊂ B and compact in H.