$L^p$-asymptotic stability of 1D damped wave equations with localized and linear damping

Kafnemer, Meryem; Mebkhout, Benmiloud; Jean, Frédéric; Chitour, Yacine

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 1

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 ${\left(S_p\left(t\right)\right)}_{t\ge 0}$ associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether $p\ge 2$ or $p<2$.

Classification : 93D20, 35L05
Keywords: Linear, 1D wave, localized, damping, $L^p$ asymptotic stability

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