Precise dispersive estimates for the wave equation inside cylindrical convex domains

Abstract

In this work, we establish precise local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains Ω ⊂ R 3 \Omega \subset \mathbb {R}^3 with smooth boundary ∂ Ω ≠ ∅ \partial \Omega \neq \emptyset . This result is the improved estimates established by Len Meas [C. R. Math. Acad. Sci. Paris 355 (2017), pp. 161–165]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain Ω \Omega have been proved by Blair, Smith, Sogge [Proc. Amer. Math. Soc. 136 (2008), pp. 247–256; Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), pp.1817–1829]. Optimal estimates in strictly convex domains have been obtained by Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380]. Our case of cylindrical domains is an extension of the result of Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions.