Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in 𝐻 10 (Ω)×𝐻 10 (Ω) . Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor A which is bounded in 𝐻 2 (Ω) × 𝐻 2 (Ω) , where the nonlinear term f satisfies a critical exponential growth condition.


Introduction
In this paper, we study the asymptotic regularity and the longtime behaviors of the solutions for the following semilinear evolution equations: where Ω is an open-bounded set of R 3 with smooth boundary Ω,  > 0, and  satisfies Equation ( 1), which appears as a class of nonlinear evolution equations as  = 2, is used to represent the propagation problems of lengthways wave in nonlinear elastic rods and ion-sonic of space transformation by weak nonlinear effect (see, for instance, [1][2][3][4]): and as  = 3 (named the Karman equation) is used to represent the flow of condensability airs in the across velocity of sound district (see, for instance, [5]) In [6], the authors have discussed the existence of global solutions in  1 0 (Ω) ×  1 0 (Ω) under the assumptions that the initial values are sufficiently small.In [7,8], the authors have discussed the nonexistence of global week solutions for the following system: (  )  +  +  (,   ) =  () , (5) where , , and  are nonlinear operators.As  = 2, in [9], the authors have discussed the long-time behaviors of solutions of (1) in  1 0 (Ω) ×  1 0 (Ω); specifically in [10], the authors have discussed the long-time behaviors of solutions of (1) in  1, 0 (Ω) ×  1, 0 (Ω).However, an open question remains whether the global attractor regularizes in the critical case and as 3 ≤  ≤ 6, the long-time behaviors of solutions of (1) have not been considered completely up to now.In this paper, we try to discuss the problem.
In the study of the global attractor regularization, the critical nonlinearity exponent brings a difficulty.About the regularity of attractor for the strongly damped wave equations, for the subcritical case, the authors in [11] have proved that the global attractor is bounded in  2 × 1 .For the critical case, Pata and Zelik [12] have proved that the global attractor is bounded in  2 ×  2 when the nonlinearity (⋅) satisfies lim inf || → ∞ (  ()/) > − 1 , for all  ∈ R, and the authors also pointed out further that one can prove the regularity of the attractor when (⋅) only satisfies the natural assumptions ISRN Mathematical Analysis (which have been realized recently in [13,14]).A general way is to obtain higher regularity of the solutions than their initial values (see, for instance, [11,12]).Then we can get the global attractor regularization.However, since (1) contains terms (|  | −2   )  ( > 2) and Δ  , they are essentially different from usual strongly damped wave equations and it is very difficult for us to obtain better regularity of the solutions of (1).Therefore, we must propose a new and more general way to study the smoothing of global attractors for (1).
In this paper, we will apply the techniques introduced in Zelik [15], Sun and Yang [13], and Yang and Sun [14] to overcome the difficulty due to the critical nonlinearity and establish the asymptotic regularity of solutions.Based on this regularity result, we obtain asymptotically compactness of the semigroup {()} ≥0 in  1 0 (Ω) ×  1 0 (Ω), and the existence of the compact attractor A has been proved.Moreover the compact attractor A is bounded in  2 (Ω) ×  2 (Ω).

Functional Setting
In what follows, we give some notations which will be used throughout this paper.Let Ω be a bounded subset of R 3 with a sufficiently smooth boundary Ω,  =  1 0 (Ω) and 1/2 , and positive operator on  2 (Ω) defined by For the family of Hilbert spaces ( /2 ),  ∈ R, their inner products and norms are, respectively, Then we have the continuous embedding ( The product Hilbert spaces endowed with the usual inner products and norms Denote by  any positive constant, which may be different from line to line and even in the same line. Throughout the paper we assume that the function  ∈ C 1 (R, R) satisfies the following conditions: Also, let  admit the decomposition  =  0 +  1 , where 0 ()  ≥ 0, ∀ ∈ R, lim sup where  1 is the first eigenvalue of −Δ in  1 0 (Ω) with the Dirichlet boundary condition.Without loss of generality, we can think  is large enough, say,  ≥ 3. Notice that by (13) and (15), there exists  <  1 , such that We denote by  the function which is easily seen to satisfy the inequalities We will complete our task exploiting the transitivity property of exponential attraction [16,Theorem 5.1], which we recall below for the readers' convenience.
Lemma 1 (see [16]).Let K 1 , K 2 , and K 3 be subsets of  0 such that for some V 0 ≥ 0 and some  0 ≥ 0. Then it follows that

Global Solution in 𝐻
Exploiting the dissipative conditions ( 16) and (18) and the standard energy estimates technique, it is easy to obtain the following result.
Lemma 2. Assume that     ( 0 ,  1 )    0 <  (23) for some  > 0. Then one has the following estimates: ( Combining with (11), we are led to the following estimation: Applying (i), estimation (ii) follows by denoting Λ 2 = (Λ 1 + |()| 2  2 + mes(Ω)).Based on the estimations above, the existence and uniqueness of the global weak solutions for (1) with the initial conditions can be obtained by standard Faedo-Galerkin method, which we omit here (see examples in Evans [17]).Theorem 3. Let Ω ⊂ R 3 be a bounded domain.The assumptions (11) The weak solutions of (1) are unique and continuously dependent on initial conditions.
From Theorem 3 and Lemma 4, the initial boundary value problem ( 1) is equivalent to a continuous semigroup {()} ≥0 defined by

Bounded Absorbing Set
We now deal with the dissipative feature of the semigroup {()} ≥0 .Namely, we show that the trajectories originating from any given bounded set eventually fall, uniformly in time, into a bounded absorbing set  0 ⊂  0 .
Proof.Multiplying (1) by   (),   , respectively, integrating in  over Ω, and then integrating in  on [, ] and associated with Theorem 5 and Sobolev embedding  1 0 →   , we can get the conclusions above.
Hereafter, we always assume that for some  > 0 and V =   + , is the bounded absorbing set of {()} ≥0 in  0 obtained in Theorem 5.

Asymptotic Regularity of the Solutions
In this section, we will establish some a priori estimates about the solutions of (1), which are the basis of our analysis.Let (, ) be a unique weak solution of (1) corresponding to the initial data  0 = ( 0 ,  1 ) ∈  1 0 (Ω) ×  1 0 (Ω).We decompose  into the sum where V() and () are the solutions to the problems It is convenient to denote Lemma 7.For any  ≥ 0, there exist  2 =  2 () ≥ 0 and  0 > 0, such that whenever ‖ 0 ‖ 0 ≤ , it follows that the constant  0 is independent of  and .
Remark 10.From the proof of Lemma 9, we observe that the decomposition V 1 () and  1 () also satisfy further that In what follows we begin to establish the asymptotic regularity of the solutions.Now we can claim the following result. then here where Combining with ( 2), (63), and ( − 2)/(4 − 2) ≤ 1, we have Using Lemma 8 to deal with the nonlinear term, we get Using Remark 10, we have Moreover, from Lemma 2, Remark 10, and Lemma 8, there exists a constant  4 =  4 () such that, for all  ≥ 0,       () where  = min(( − )/2, 2).Using the Gronwall inequality and integrating over [0, ], we get that there exists a constant   (> 0) which depends only on the   -bounds of   such that, for any  ≥ 0 and  0 ∈   , In the following, based on Lemmas 11 and 12, we perform a bootstrap argument, whose proof is similar to that of Lemmas 11 and 12 (e.g., see [11,13]).Here we only point out the results and omit the proof.
where   0 depends on the  0 -bounds of   (determined in Lemma 7).

Global Attractor
Collecting now Theorem 5, Lemmas 7 and 8, and Theorem 15, we establish that {()} ≥0 is asymptotically compact.Therefore, by means of well-known results of the theory of dynamical systems we get the following.