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In this paper we study the stability of the unique continuation in the caseof the wave equation with variable coefficients independent of time. We prove alogarithmic estimate in a arbitrary domain of ${\mathbb R}^{n+1}$, where allthe parameters are calculated explicitly in terms of the $C^1$-norm of thecoefficients and on the other geometric properties of the problem. We use theCarleman-type estimate proved by Tataru in 1995 and an iteration for localsstability. We apply the result to the case of a wave equation with data on acylinder an we get a stable estimate for any positive time, also after thefirst conjugate point for the geodesics of the metric related to the variablecoefficients.

Stability of the unique continuation for the wave operator via Tataru inequality and applications