Abstract
In this paper, we study the Lp-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 (Sp(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.
Highlights
This paper is concerned with the asymptotic stability of the one dimensional wave equation with a localized damping term and Dirichlet boundary conditions
The function a is a continuous non-negative function on [0, 1], bounded from below by a positive constant on some non-empty open interval ω of (0, 1), which represents the region of the domain where the damping term is active
Stability results are proved under a geometric condition imposed on the damping domain ω: it is properly introduced in the early work [18] where the semi-linear problem is considered even in higher dimension and the geometric condition is extended and characterized in [13]
Summary
This paper is concerned with the asymptotic stability of the one dimensional wave equation with a localized damping term and Dirichlet boundary conditions. As for more general functional frameworks, in particular Lp-based spaces with p = 2, few results exist and one reason is probably due to the fact that, in such non-Hilbertian framework, the semi-group associated with the d’Alembertian (i.e., the linear operator defining the wave equation) is not defined in general as soon as the space dimension is larger than or equal to two, see e.g., [16]). This is why most of the existing results focus on several stabilization issues only in one spatial dimension.
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