Almost sure existence of global solutions for supercritical semilinear wave equations

We prove that for almost every initial data (u(0), u(1)) is an element of H-s x Hs-1 with s > p-3/p-1 there exists a global weak solution to the supercritical semilinear wave equation partial derivative(2)(t)u - Delta u + vertical bar u vertical bar(p-1)u = 0 where p > 5, in both R-3 and T-3. This improves in a probabilistic framework the classical result of Strauss [20] who proved global existence of weak solutions associated to H-1 x L-2 initial data. The proof relies on techniques introduced by Oh and Pocovnicu in [16] based on the pioneer work of Burq and Tzvetkov in [7]. We also improve the global well-posedness result in [21] for the subcritical regime p < 5 to the endpoint s = P-3/p-1. (C) 2020 Elsevier Inc. All rights reserved.