Non-convergence and convergence of bounded solutions to semilinear wave equation with dissipative boundary condition

TL;DR

This paper investigates the long-term behavior of solutions to a semilinear wave equation with dissipative boundary conditions, revealing conditions for convergence or non-convergence to equilibrium states.

Contribution

It analyzes the set of equilibria, demonstrates non-stabilizing solutions for certain nonlinear sources, and establishes convergence rates when the source is a { extltilde}ojasiewicz-type function.

Findings

Set of equilibria can be infinite.

Solutions may not stabilize but approach a continuum of states.

Convergence rate depends on the { extltilde}ojasiewicz exponent.

Abstract

This paper is concerned with the long-time dynamics of semilinear wave equation subject to dissipative boundary condition. To do so, we first analyze the set of equilibria, and show it could contain infinitely many elements. Second, we show that, for some nonlinear interior sources, the wave equations have solutions that do not stabilize to any single function, while they approach a continuum of such functions. Finally, if the interior source is a {\L}ojasiewicz-type function, the solution of the wave equation converges to an equilibrium at a rate that depends on the {\L}ojasiewicz exponent, although the set of equilibria is infinite.