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Journal d'Analyse Mathématique

, Volume 133, Issue 1, pp 91–131 | Cite as

Large global solutions for energy supercritical nonlinear wave equations on ℝ3+1

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Abstract

For the radial energy-supercritical nonlinear wave equation □u = −u tt + Δu = ±u 7 on R3+1, we prove the existence of a class of global in forward time C -smooth solutions with infinite critical Sobolev norm 7/6(R3 1/6(R3). These solutions admit a precise asymptotic description and are stable under suitably small perturbations. We also show that for the defocussing energy supercritical wave equation, we can construct such solutions which moreover satisfy the size condition \({\left\| {u\left( {0,.} \right)} \right\|_{L_x^\infty \left( {\left| x \right| \geqslant 1} \right)}} > M\) for arbitrarily prescribed M > 0. These solutions are stable under suitably small perturbations and admit a precise asymptotic description. Also, these solutions experience infinite inflation of the critical 7/6-norm in any forward light cone. Our method proceeds by regularization of self-similar solutions which are smooth away from the light-cone but singular on the light-cone. The argument crucially depends on the supercritical nature of the equation. Our approach should be seen as part of the program initiated in [10], [11], [4].

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Copyright information

© Hebrew University Magnes Press 2017

Authors and Affiliations

  1. 1.Bâtiment des MathématiquesEPFL, Station 8LausanneSwitzerland
  2. 2.Department of MathematicsThe University of ChicagoChicagoUSA

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