Damped wave equation. (Equation des ondes amorties.) (French) Zbl 0863.58068

Lebeau, G.

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].

Reviewer: P.Godin (Bruxelles)

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

Keywords:

wave equation; stabilization; Riemannian manifold; Laplacian; evolution problem; spectrum; geodesics