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

About this Title

Charles Collot, Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis, 28 avenue Valrose, France

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
MSC: Primary primary, 35B44, secondary, 35L05, 58B99

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Table of Contents

Chapters

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

Abstract

We consider the semilinear focusing wave equation \[ \partial _{tt}u-\Delta u-u|u|^{p-1}=0\] in large dimensions $d\geq 11$ and in the radial case. For a range of energy supercritical nonlinearities $p>p(d)>1+\frac {4}{d-2}$, for each integer large enough $\ell >\alpha (d,p)>2$, we construct a Lipschitz manifold of codimension $\ell -1$ of solutions blowing up in finite time $T$ by concentrating the soliton (stationnary state) profile: \[ u(t,r)\sim \frac {1}{\lambda (t)^{\frac 2{p-1}}}Q\left (\frac {r}{\lambda (t)}\right )\] at the quantized blow up rate : \[ \lambda (t)\sim c_u(T-t)^{\frac \ell \alpha }.\] The solutions can be chosen $C^{\infty }$ and compactly supported. In that case the blow up is of type II i.e all norms below scaling remain bounded \[ \limsup _{t\uparrow T}\|\nabla ^su(t),\nabla ^{s-1}\partial _tu(t)\|_{L^2}<+\infty \ \ \mbox {for}\ \ 1\leq s<s_c=\frac {d}{2}-\frac {2}{p-1}.\] Our 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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