Stabilization of the nonlinear damped wave equation via linear weak observability

TL;DR

This paper establishes sharp energy decay rates for nonlinear damped wave equations using weak observability estimates, combining convexity methods and extending results to nonlinear systems.

Contribution

It introduces a novel approach that combines weak observability with convexity methods to analyze energy decay in nonlinear damped systems.

Findings

Proves sharp decay rates for nonlinear damped wave equations.

Extends weak stabilization results to nonlinear systems.

Provides a new framework combining observability and convexity techniques.

Abstract

We consider the problem of energy decay rates for nonlinearly damped abstract infinite dimensional systems. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely a weak observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine the optimal-weight convexity method of Alabau-Boussouira and a methodology of Ammari-Tucsnak for weak stabilization by observability. Our results extend to nonlinearly damped systems, those of Ammari and Tucsnak. At the end, we give an appendix on the weak stabilization of linear evolution systems.