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 $\Omega \subset \mathbb {R}^3$ with smooth boundary $\partial \Omega \neq \emptyset$. 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 $\Omega$ 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.