Dispersive estimates for the wave equation inside cylindrical convex domains: A model case

In this work, we will establish local in time dispersive estimates for solutions to the model-case Dirichlet wave equation inside a cylindrical convex domain $\mathrm{\Omega }?{\mathbb{R}}^3$ with a smooth boundary $?\mathrm{\Omega }\ne ?$. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Nonoptimal Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair–Smith–Sogge 1], 2]. Better estimates in strictly convex domains have been obtained in 4]. Our case of cylindrical domains is an extension of the result of 4] in the case where the curvature radius ≥0 depends on the incident angle and vanishes in some directions.