Almost Sure Existence of Global Solutions for supercritical Semilinear Wave Equations

Mickaël Latocca (1)

1 DMA - Département de Mathématiques et Applications - ENS Paris

We prove that for almost every initial data (u0, u1) ∈ H s × H s−1 with s > p−3 p−1 there exists a global weak solution to the supercritical semilinear wave equation ∂ 2 t u − ∆u + |u| 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 [16] who proved global existence of weak solutions associated to H 1 × L 2 initial data. The proof relies on techniques introduced by T. Oh and O. Pocovnicu in [13] based on the pioneer work of N. Burq and N. Tzvetkov in [5]. We also improve the global well-posedness result in [17] for the subcritical regime p < 5 to the endpoint s = p−3 p−1. Acknowledgement The author warmly thanks Nicolas Burq and Isabelle Gallagher for careful advising during this work and suggesting this problem.

Domaines: Equations aux dérivées partielles [math.AP]

Mickaël Latocca. Almost Sure Existence of Global Solutions for supercritical Semilinear Wave Equations. 2018. ⟨hal-01876990⟩