A note on a strongly damped wave equation with fast growing nonlinearities

TL;DR

This paper proves the global well-posedness and existence of a global attractor for a strongly damped wave equation with nonlinearities of arbitrary polynomial growth, without requiring a global Lyapunov function.

Contribution

It establishes well-posedness and attractor existence for a nonlinear damped wave equation with fast-growing nonlinearities under standard dissipativity conditions, without needing a global Lyapunov function.

Findings

The initial boundary value problem is globally well-posed.

The semigroup has a global attractor in the phase space.

Results hold for nonlinearities with arbitrary polynomial growth.

Abstract

A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved the initial boundary value problem for the considered equation is globally well-posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function.