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Damped wave equation. (Equation des ondes amorties.) (French) Zbl 0863.58068
Boutet de Monvel, Anne (ed.) et al., Algebraic and geometric methods in mathematical physics. Proceedings of the 1st Ukrainian-French-Romanian summer school, Kaciveli, Ukraine, September 1-14, 1993. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 19, 73-109 (1996).
Let $$(M,g)$$ be a $$C^\infty$$ compact Riemannian manifold with boundary, with Laplacian $$\Delta$$, and let $$a$$ be a $$C^\infty (M, \mathbb{R}^+)$$ function. One considers the evolution problem $\begin{cases} \bigl(\partial^2_t - \Delta+ 2a(x) \partial_t\bigr) u=0 \text{ in } \mathbb{R}_t \times M, u=0 \text{ on } \mathbb{R}_t \times \partial M, \\ u|_{t=0} = u_0\in H^1_0 (M), \quad {\partial u \over \partial t} |_{t=0} = u_1\in L^2(M).\end{cases} \tag{*}$ The author obtains sharp estimates for the resolvent of $$A_a = \left(\begin{smallmatrix} 0 & Id \\ \Delta & -2a \end{smallmatrix} \right)$$ and for the energy. The best exponential decay rate of the solutions of the evolution problem (*) is computed in terms of the spectrum and of the average of $$a(x)$$ on the geodesics of $$M$$.
For the entire collection see [Zbl 0833.00031].

##### MSC:
 58J45 Hyperbolic equations on manifolds 35L05 Wave equation 35S15 Boundary value problems for PDEs with pseudodifferential operators 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C22 Geodesics in global differential geometry