The main results of the present paper consist in some quantitative estimates for solutions to the wave equation ∂t2u-div(A(x)∇xu)=0\documentclass[12pt]{minimal}
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\begin{document}$$\partial ^2_{t}u-\text{ div }(A(x)\nabla _x u)=0$$\end{document}. Such estimates imply the following strong unique continuation properties: (a) if u is a solution to the the wave equation and u is flat on a segment {x0}×J\documentclass[12pt]{minimal}
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\begin{document}$$\{x_0\}\times J$$\end{document} on the t axis, then u vanishes in a neighborhood of {x0}×J\documentclass[12pt]{minimal}
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\begin{document}$$\{x_0\}\times J$$\end{document}. (b) Let u be a solution of the above wave equation in Ω×J\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \times J$$\end{document} that vanishes on a a portion Z×J\documentclass[12pt]{minimal}
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\begin{document}$$Z\times J$$\end{document} where Z is a portion of ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega $$\end{document} and u is flat on a segment {x0}×J\documentclass[12pt]{minimal}
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\begin{document}$$\{x_0\}\times J$$\end{document}, x0∈Z\documentclass[12pt]{minimal}
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\begin{document}$$x_0\in Z$$\end{document}, then u vanishes in a neighborhood of {x0}×J\documentclass[12pt]{minimal}
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\begin{document}$$\{x_0\}\times J$$\end{document}. The property (a) has been proved by Lebeau (Commun Partial Differ Equ 24:777–783, 1999).
