# Global logarithmic stability of a Cauchy problem for anisotropic wave equations

## TL;DR

This paper establishes logarithmic stability estimates for the Cauchy problem of anisotropic wave equations, transferring known elliptic stability results via a specialized transform, and also proves uniqueness of continuation from Cauchy data.

## Contribution

It introduces a method to derive stability estimates for anisotropic wave equations using a Fourier-Bros-Iagolnitzer transform, extending elliptic results to wave equations.

## Key findings

- Logarithmic stability estimates for anisotropic wave equations.
- Uniqueness of continuation from Cauchy data.
- Control of residual terms via the transform.

## Abstract

We discuss the 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 uniqueness of solutions of an anisotropic wave equation. In the case where the uniqueness holds, the natural problem that arise in this context is to estimate the solutions, in some appropriate space, in terms of the Cauchy data. We aim in this paper to transfer, via a reduced Fourrier-Bros-Iagolnitzer transform, the known stability estimates for the Cauchy problem for elliptic equations to that for waves equations. By proceeding in that way the main difficulty is to control the residual terms, induced by the reduced Fourrier-Bros-Iagolnitzer transform, by the Cauchy data. Also, the uniqueness of continuation from Cauchy data is obtained as byproduct of stability estimates.