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Random data Cauchy theory for supercritical wave equations II: a global existence result

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Abstract

We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in \(\bigcap_{s<1/2} (H^s(\Theta)\times H^{s-1}(\Theta))\). We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations, inspired by earlier works by Bourgain [2, 3] on the non linear Schrödinger equation, which allow us to obtain also precise large time dynamical informations on our solutions.

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris XI, 91 405, Orsay Cedex, France

    Nicolas Burq

  2. Institut Universitaire de France, Paris, France

    Nicolas Burq

  3. Département de Mathématiques, Université Lille I, 59 655, Villeneuve d’Ascq Cedex, France

    Nikolay Tzvetkov

Authors
  1. Nicolas Burq
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  2. Nikolay Tzvetkov
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Correspondence to Nicolas Burq.

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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 II: a global existence result. Invent. math. 173, 477–496 (2008). https://doi.org/10.1007/s00222-008-0123-0

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  • DOI: https://doi.org/10.1007/s00222-008-0123-0

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