Global attractors for wave equations with nonlinear interior damping and critical exponents

Abstract

In this paper we study the global attractors for wave equations with nonlinear interior damping. We prove the existence, regularity and finite dimensionality of the global attractors without assuming a large value for the damping parameter, when the growth of the nonlinear terms is critical.

Semilinear damped wave equation in locally uniform spaces

“…In contrast to the bounded domain case, the energy of the solutions does not necessarily decrease over time. This may be attributed to the last element in (12) and the absence of the embeddings between the weighted spaces of the same weight. Thus, it seems that an additional assumption has to be made in order to show any dissipation of energy.…”

“…The finite dimensionality has been discussed in [17] and has been achieved even for critical nonlinearities in [4] and [12]. In the case of linear damping, the results include the existence of a global attractor on manifolds [11], asymptotic regularity of solutions of perturbed damped wave equation with nonlinearity of arbitrary growth [21] and recently in [10] the existence and smoothness of a global attractor for the equation with critical nonlinearity in d = 3.…”

The global attractor for a suspension bridge with memory and partially hinged boundary conditions

“…Here we show that the semigroup S(t) is asymptotically smooth in H (cf. [5,11]). First, we recall the following results.…”

Global attractor for hyperbolic equation with nonlinear damping and linear memory

“…In this section, we will first give some a priori estimates about the energy inequalities on account of the idea presented in [1,2,4,5]. Then we use Theorem 4.1 to establish the asymptotic compactness in H. For convenience, we always denote by B 1 a bounded absorbing set obtained in Lemma 3.1.…”

“…The following process is derived from the standard energy method given in [1,2,4,5]. Let (u i (t), u it (t), η t i (s)) (i = 1, 2) be the corresponding solution to (u i 0 , v i 0 , η i 0 ) ∈ B 1 and let w(t) = u 1 (t) − u 2 (t) and ζ t (s) = η t 1 (s) − η t 2 (s).…”