Abstract
In this paper, we address an inverse problem for the Korteweg-de Vries equation posed on a bounded domain with boundary conditions proposed by Colin and Ghidaglia. More precisely, we retrieve the principal coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability of this inverse problem relies on a Carleman estimate for the linearized Korteweg-de Vries equation and the Bukhgeı̌m-Klibanov method.
Highlights
1 Introduction This paper is concerned with the Korteweg-de Vries (KdV) equation with a non-constant coefficient posed on a finite interval
Based on the results in [ – ], if h = h(x) is the function describing the variations in depth of the channel, the KdV equation becomes
Stability estimates play a special role in the theory of inverse problems of mathematical physics that are ill-posed in the classical sense
Summary
This paper is concerned with the Korteweg-de Vries (KdV) equation with a non-constant coefficient posed on a finite interval. Stability estimates play a special role in the theory of inverse problems of mathematical physics that are ill-posed in the classical sense. They determine the choice of regularization parameters and the rate at which solutions of regularized problems converge to an exact solution. To the best of our knowledge, the only result in the literature concerning the determination of coefficients for the KdV equation is in [ ], where the author considered the KdV equation with boundary conditions as in Let a , α and K be given positive constants, y ∈ W and a, a ∈ (a , α)
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