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In this note we consider a strictly convex domain Ω ⊂ Rd of dimension d ≥ 2 with smooth boundary ∂Ω 6= ∅ and we describe the dispersive and Strichartz estimates for the wave equation with the Dirichlet boundary condition. We obtain counterexamples to the optimal Strichartz estimates of the flat case; we also discuss the some results concerning the dispersive estimates.

Dispersive and Strichartz estimates for the wave equation in domains with boundary