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Article Dans Une Revue Journal of Differential Equations Année : 2021

Almost sure existence of global solutions for supercritical semilinear wave equations

Résumé

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 .
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Dates et versions

hal-04196609 , version 1 (05-09-2023)

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Mickaël Latocca. Almost sure existence of global solutions for supercritical semilinear wave equations. Journal of Differential Equations, 2021, 273, pp.83-121. ⟨10.1016/j.jde.2020.11.045⟩. ⟨hal-04196609⟩
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