Lp-asymptotic stability of 1D damped wave equations with localized and linear damping

TL;DR

This paper investigates the exponential stability of one-dimensional linear damped wave equations with localized linear damping in L^p spaces, demonstrating well-posedness and stability for all p in (1,∞).

Contribution

It establishes L^p-asymptotic stability for the 1D damped wave equation with arbitrary localized linear damping, extending stability results across all p in (1,∞).

Findings

Proves well-posedness of the associated semigroup.

Establishes exponential stability for all p in (1,∞).

Uses multiplier method depending on p range.

Abstract

In this paper, we study the $L^p$-asymptotic stability of the one-dimensional linear damped wave equation with Dirichlet boundary conditions in $[0,1]$, with $p\in (1,\infty)$. The damping term is assumed to be linear and localized to an arbitrary open sub-interval of $[0,1]$. We prove that the semi-group $(S_p(t))_{t\geq 0}$ associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether $p\geq 2$ or $1<p<2$.