Strichartz type estimates and the wellposedness of an energy critical 2D wave equation in a bounded domain

TL;DR

This paper establishes Strichartz estimates and analyzes the well-posedness of an energy-critical 2D wave equation with boundary conditions, demonstrating global existence below a threshold and instability above it.

Contribution

It introduces a new Strichartz estimate for 2D wave equations on bounded domains and applies it to prove well-posedness and instability results for the critical nonlinear wave equation.

Findings

Global well-posedness below the energy threshold

Instability in the supercritical case

Spectral projector estimates for the Laplacian

Abstract

We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded 2D domain {\Omega}. First, we prove an appropriate Strichartz type estimate using the $L^q$ spectral projector estimates of the Laplace operator. Our proof follows Burq-Lebeau-Planchon [5]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser-Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data.