Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

TL;DR

This paper investigates the long-term behavior of solutions to a 3-D strongly damped wave equation with displacement-dependent damping and a critical nonlinear source, establishing the existence and regularity of global attractors.

Contribution

It proves the existence of a global attractor for the equation's semigroup and shows its boundedness and regularity in higher Sobolev spaces.

Findings

Existence of a global attractor in H_{0}^{1}(\Omega) imes L_{2}(\Omega)

Global attractor is bounded in H^{2}(\Omega) imes H^{2}(\Omega)

Global attractor exists in H^{2}(\Omega)igcap H_{0}^{1}(\Omega) imes H_{0}^{1}(\Omega)

Abstract

In this paper the long time behaviour of the solutions of 3-D strongly damped wave equation is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H_{0}^{1}(\Omega)\times L_{2}(\Omega) and then it is proved that this global attractor is a bounded subset of H^{2}(\Omega)\times H^{2}(\Omega) and also a global attractor in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega).