Almost Sure Existence of Global Solutions for Supercritical Semilinear Wave Equations

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 ${\partial }_t^{2}u-\Delta u+|u{|}^{p-1}u=0$ 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 $s=\frac{p-3}{p-1}$.

Minor modifications, typos corrected, reference added, final version