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 $s=\frac{p-3}{p-1}$.

arXiv commentMinor modifications, typos corrected, reference added, final version

References in corpus (4)

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• Random data Cauchy theory for supercritical wave equations II : A global existence result
• Almost sure global well posedness for the radial nonlinear Schrodinger equation on the unit ball I: the 2D case
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Cited by in corpus (3)

• Concerning the pathological set in the context of probabilistic well-posedness
• Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus
• Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation