Abstract
We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H s(M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H s(M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary.
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Mathematics Subject Classification (2000)
35Q55, 35BXX, 37K05, 37L50, 81Q20
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Burq, N., Tzvetkov, N. Random data Cauchy theory for supercritical wave equations I: local theory. Invent. math. 173, 449–475 (2008). https://doi.org/10.1007/s00222-008-0124-z
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DOI: https://doi.org/10.1007/s00222-008-0124-z