Decay of solutions of the elastic wave equation with a localized dissipation
The author studies two classical elastic wave equations: \(\partial_t^2 u-\Delta_e u=0\) and its damped version \(\partial_t^2 u-\Delta_e u+a(x)\partial_t u=0\) posed on a Riemannian manifold with a smooth and compact boundary and supplemented with damped and Dirichlet boundary conditions, respectively. Here, \(\Delta_e=\mu\Delta+(\mu+\lambda)\nabla ({div})\) where \(\mu>0\), \(2\mu+\lambda>0\), and \(\Delta\), \(\nabla\) and \(div\) are the Laplacian, gradient and the divergence operator associated to the Riemannian metric. For those initial-boundary value problem, the logarithmic decay in time of the energy functional is shown.
scientific article; zbMATH DE number 2109722
scientific article