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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  ∈ (1,  ∞ ). 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 ≥0  associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether  p  ≥ 2 or 1 &lt;  p  &lt; 2.

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

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