Random data Cauchy theory for supercritical wave equations II : A global existence result

TL;DR

This paper establishes the global existence of strong solutions for a supercritical wave equation on a three-dimensional ball with random initial data, utilizing a new random data Cauchy theory and invariant measures.

Contribution

It introduces a novel global existence result for supercritical wave equations with random initial data in three dimensions, extending previous theoretical frameworks.

Findings

Global strong solutions exist for a large set of random supercritical initial data.

The paper develops a general random data Cauchy theory for supercritical wave equations.

Provides large time dynamical information about solutions using invariant measures.

Abstract

We prove that the subquartic wave equation on the three dimensional ball $\Theta$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\cap_{s<1/2} H^s(\Theta)$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work \cite{BT2} and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions.