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

TL;DR

This paper analyzes wave equation dispersion inside strictly convex domains, revealing a specific decay rate loss caused by boundary-induced singularities, with implications for understanding wave behavior in such geometries.

Contribution

It provides the first detailed description of dispersion and decay rates for the wave equation in strictly convex domains with boundary effects.

Findings

Optimal fixed time decay rate established

Boundary singularities cause a t^{1/4} loss in decay

Swallowtail singularities influence wave front set behavior

Abstract

We consider a model case for a strictly convex domain of dimension $d\geq 2$ with smooth boundary 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.