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

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 is an element of (1, infinity). 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 < p < 2.