Dispersive estimates for the wave equation inside cylindrical convex domains

Abstract

The dispersive and Strichartz estimates are essential for establishing well posedness results for nonlinear equations as well as long time behaviour of solutions to the equation. While in the boundary-less case these estimates are well understood, the case of boundary the situation can become much more difficult. In this work, we establish local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains \(\Omega\subset\mathbb{R}^3\) with smooth boundary \(\partial\Omega\neq \emptyset\). In this paper, we provide detailed proofs of the results established in [16, 17]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain \(\Omega\) have been proved by Blair, Smith, Sogge [4, 5]. Optimal estimates in strictly convex domains have been obtained in [12]. Our case of cylindrical domains is an extension of the result of [12] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions.