Almost Sure Existence of Global Solutions for Supercritical Semilinear Wave Equations

TL;DR

This paper demonstrates that for almost all initial data in a certain Sobolev space, there exist global weak solutions to supercritical semilinear wave equations, extending classical results through probabilistic methods.

Contribution

It establishes almost sure global existence of solutions for supercritical wave equations in a probabilistic framework, improving previous deterministic results.

Findings

Almost sure global solutions exist for initial data in $H^s \times H^{s-1}$ with $s > \frac{p-3}{p-1}$.

Improves classical results by Strauss in a probabilistic setting.

Extends global well-posedness to the endpoint regularity for subcritical cases.

Abstract

We prove that for almost every initial data $(u_0,u_1) \in H^s \times H^{s-1}$ with $s > \frac{p-3}{p-1}$ there exists a global weak solution to the supercritical semilinear wave equation $\partial _t^2u - \Delta u +|u|^{p-1}u=0$ where $p>5$, in both $\mathbb{R}^3$ and $\mathbb{T}^3$. This improves in a probabilistic framework the classical result of Strauss who proved global existence of weak solutions associated to $H^1 \times L^2$ initial data. The proof relies on techniques introduced by T. Oh and O. Pocovnicu based on the pioneer work of N. Burq and N. Tzvetkov. We also improve the global well-posedness result of C. Sun and B. Xia for the subcritical regime $p<5$ to the endpoint $s=\frac{p-3}{p-1}$.