# Dispersive estimates for the Schrödinger equation in a model convex domain and applications

### Oana Ivanovici

Sorbonne University, Paris, France

## Abstract

We consider an anisotropic model case for a strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq 2$ with smooth boundary $\partial\Omega\neq\emptyset$ and we describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semiclassical flow: a loss of $(\frac ht)^{1/4}$ occurs with respect to the boundaryless case due to repeated swallowtail-type singularities, and is proven optimal. Corresponding Strichartz estimates allow us to solve the cubic nonlinear Schrödinger equation on such a three-dimensional model convex domain, hence matching known results on generic compact boundaryless manifolds.

## Cite this article

Oana Ivanovici, Dispersive estimates for the Schrödinger equation in a model convex domain and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 40 (2023), no. 4, pp. 959–1008

DOI 10.4171/AIHPC/75