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Type II Blow up Manifolds for the Energy Supercritical Semilinear Wave Equation


About this Title

Charles Collot

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 252, Number 1205
ISBNs: 978-1-4704-2813-6 (print); 978-1-4704-4379-5 (online)
DOI: https://doi.org/10.1090/memo/1205
Published electronically: January 29, 2018
Keywords:blow up \* concentration \* manifold \* soliton \* wave equation

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. The Linearized Dynamics and the Construction of the Approximate Blow up Profile
  • Chapter 3. The Trapped Regime
  • Chapter 4. End of the Proof
  • Chapter 5. Lipschitz Aspect and Codimension of the Set of Solutions Described by Proposition 3.2
  • Appendix A. Properties of the Stationary State
  • Appendix B. Equivalence of Norms
  • Appendix C. Hardy Inequalities
  • Appendix D. Coercivity of the Adapted Norms
  • Appendix E. Specific Bounds for the Analysis

Abstract


We consider the semilinear focusing wave equationin large dimensions and in the radial case. For a range of energy supercritical nonlinearities , for each integer large enough , we construct a Lipschitz manifold of codimension of solutions blowing up in finite time by concentrating the soliton (stationnary state) profile:at the quantized blow up rate :The solutions can be chosen and compactly supported. In that case the blow up is of type II i.e all norms below scaling remain boundedOur analysis adapts the robust energy method developed for the study of energy critical bubbles by Merle-Raphaël-Rodnianski, Raphaël-Rodnianski and Raphaël-Schweyer, the study of this issue for the supercritical semilinear heat equation done by Herrero-Velázquez, Matano-Merle and Mizoguchi, and the analogous result for the energy supercritical Schrödinger equation by Merle-Raphaël-Rodnianski.

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