Global logarithmic stability of a Cauchy problem for anisotropic wave equations

Mourad Bellassoued () and Mourad Choulli ()
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Mourad Bellassoued: University Tunis El Manar, ENIT-LAMSIN
Mourad Choulli: Université de Lorraine

Partial Differential Equations and Applications, 2023, vol. 4, issue 3, 1-44

Abstract: Abstract We discuss a Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee the uniqueness of continuation of solutions of an anisotropic wave equation. In the case where the uniqueness holds, the natural problem that arise naturally in this context is to estimate the solutions, in some appropriate space, in terms of norms of the Cauchy data. We aim in this paper to convert, via a reduced Fourier–Bros–Iagolnitzer transform, the known stability of the Cauchy problem for anisotropic elliptic equations to stability of a Cauchy problem for anisotropic waves equations. By proceeding in that way one the main difficulties is to control the residual terms, induced by the reduced Fourier–Bros–Iagolnitzer transform, by a Cauchy data. Also, a uniqueness of continuation result, from Cauchy data, is obtained as byproduct of stability results.

Keywords: Anisotropic wave equations; Cauchy problem; Uniqueness of continuation; Reduced Fourier–Bros–Iagolnitzer transform; Stability estimate; 35B60 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00242-2

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