Local energy decay and diffusive phenomenon in a dissipative wave guide

Abstract

We prove the local energy decay for the wave equation in a wave guide with dissipation at the boundary. It appears that for large times the dissipated wave behaves like a solution of a heat equation in the unbounded directions. The proof is based on resolvent estimates, in particular at the low and high frequency limits. Since the eigenvectors for the transverse operator do not form a Riesz basis, the spectral analysis does not reduce as usual to separate subquestions on compact and Euclidean domains. One of the difficulties is then to localize the problem with respect to the non-discrete spectrum of a non-selfadjoint operator.