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 . Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor which is bounded in , where the nonlinear term satisfies a critical exponential growth condition. 1. Introduction In this paper, we study the asymptotic regularity and the long-time behaviors of the solutions for the following semilinear evolution equations: where is an open-bounded set of with smooth boundary , , and satisfies Equation (1), which appears as a class of nonlinear evolution equations as , 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, [1–4]): and as (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 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: where , , and are nonlinear operators. As , in [9], the authors have discussed the long-time behaviors of solutions of (1) in ; specifically in [10], the authors have discussed the long-time behaviors of solutions of (1) in . However, an open question remains whether the global attractor regularizes in the critical case and as , 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 . For the critical case, Pata and Zelik [12] have proved that the global attractor is bounded in when the nonlinearity satisfies , for all , and the authors also pointed out further that one can prove the regularity of the attractor when only satisfies the natural assumptions (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 and , they are essentially different from usual strongly damped