Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case

Abstract

We consider a model case for a strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq 2$ with smooth boundary $\partial\Omega\neq\emptyset$, and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a $t^{1/4}$ loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.