Decay rate estimates for the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation

TL;DR

This paper investigates the stabilization and decay rates of wave equations with subcritical nonlinearities and localized nonlinear damping, using approximation, microlocal analysis, and observability inequalities to handle complex nonlinear structures.

Contribution

It introduces a method to establish observability and decay estimates for wave equations with complex nonlinearities by truncating nonlinearities and employing microlocal analysis.

Findings

Established decay rate estimates for various nonlinear dissipative effects.

Extended known results to a broader class of nonlinear damping.

Developed a framework for analyzing wave equations with challenging nonlinear structures.

Abstract

We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be found in the lecture notes of Burq and G\'erard (2001). Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.