Exponential stability of solutions to perturbed superstable wave equations

TL;DR

This paper proves that solutions to superstable wave equations, which stabilize in finite time, remain exponentially stable under small perturbations, with added smoothing effects for higher regularity.

Contribution

It establishes exponential stability of perturbed superstable wave equations in both $L^2$ and $C^2$ spaces, including a smoothing result for $C^2$ regularity.

Findings

Solutions remain exponentially stable under small bounded perturbations.

Perturbed solutions become eventually $C^2$-smooth for any $H^1\times L^2$ initial data.

Stability results hold in both $L^2$ and $C^2$ norms.

Abstract

The paper deals with initial-boundary value problems for the linear wave equation whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in $L^2$ as well as in $C^2$ under small bounded perturbations of the wave operator. To show this for $C^2$, we prove a smoothing result implying that the solutions to the perturbed problems become eventually $C^2$-smooth for any $H^1\times L^2$-initial data.