Carleman estimate for a strongly damped wave equation and applications to an inverse problem

Abstract

In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time-dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro-differential equation with hyperbolic memory kernel., provided that u 0 , u 1 are given.Many physical phenomena are properly described by the strongly damped wave equation as (1). Such as equations of type (1) can be considered as a class of linear evolution equations governing the motion of a viscoelastic solid (for example, a bar if the space dimension N D 1 and a plate if N D 2) composed of the material of the rate type; see [1][2][3][4]. They can also be seen as field equations governing the longitudinal motion of a viscoelastic bar obeying the simply Voigt model [5]. Mathematical studies on this direct problem have been extensively developed, for example, see [1,2,[6][7][8][9] and the references therein, where the global existence and other properties of solutions are obtained. However, very little work is concerned with the inverse problem for this strongly damped wave equation.The key ingredient we follow here to determine the coefficient relies on the global Carleman estimates. For the first time, the method of Carleman estimates was introduced in the field of inverse problems in the work of Bukhgeim and Klibanov [10], also for some followup publications [11,12] of these authors. A Carleman estimate is an efficient tool not only for the unique continuation, but also for

Carleman Estimates, Observability Inequalities and Null Controllability for Interior Degenerate Non Smooth Parabolic Equations

“…Moreover, Carleman estimates are also extremely useful for several other applications, especially for unique continuation properties (for example, see [25], [36] and [40]), for inverse problems, in parabolic, hyperbolic and fractional settings, e.g. see [8], [21], [38], [48], [49], [53], [54] and their references.…”

Carleman estimates and observability inequalities for parabolic equations with interior degeneracy

“…Moreover, Carleman estimates may be a fundamental tool in inverse problems, in parabolic, hyperbolic and fractional settings, e.g. see [15], [25], [30], [31], [35], [36] and their references.…”

A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain $\Omega \times {\mathcal {O}}$ of ${\mathbb {R}}^{M+N}$

“…Starting from the pioneering works in the eighties by Bukhgeim and Klibanov (see [8,27,28] and also the monographs [7,30] and the survey papers [25,29]), Carleman estimates have been used to solve identification problems, mainly in bounded domains, associated with nondegenerate differential operators. We quote, e.g., [3,4,6,5,18,24,33,39]. On the contrary, to the best of our knowledge, Carleman estimates have not been extensively used so far in the analysis of inverse problems in unbounded domains.…”