Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Tej-eddine Ghoul, S. Ibrahim, V. T. Nguyen

Research output: Contribution to journalArticle

Abstract

We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂t 2u=∂r 2u+[Formula presented]∂ru−[Formula presented]sin⁡(2u). We construct for this equation a family of C solutions which blow up in finite time via concentration of the universal profile u(r,t)∼Q([Formula presented]), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t)∼cu(T−t)[Formula presented],ℓ∈N,ℓ>γ=γ(d)∈(1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski [49] for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

Original languageEnglish (US)
Pages (from-to)2968-3047
Number of pages80
JournalJournal of Differential Equations
Volume265
Issue number7
DOIs
StatePublished - Oct 5 2018

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Blow-up Solution
Wave equations
Nonlinear equations
Modulation
Energy
Brouwer Fixed Point Theorem
Semilinear Wave Equation
Energy Method
Stationary Solutions
Nonlinear Equations
Symmetry

Keywords

  • Blowup profile
  • Blowup solution
  • Stability
  • Wave maps

ASJC Scopus subject areas

  • Analysis

Cite this

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps. / Ghoul, Tej-eddine; Ibrahim, S.; Nguyen, V. T.

In: Journal of Differential Equations, Vol. 265, No. 7, 05.10.2018, p. 2968-3047.

Research output: Contribution to journalArticle

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abstract = "We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂t 2u=∂r 2u+[Formula presented]∂ru−[Formula presented]sin⁡(2u). We construct for this equation a family of C∞ solutions which blow up in finite time via concentration of the universal profile u(r,t)∼Q([Formula presented]), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t)∼cu(T−t)[Formula presented],ℓ∈N⁎,ℓ>γ=γ(d)∈(1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Rapha{\"e}l and Rodnianski [49] for the energy supercritical nonlinear Schr{\"o}dinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.",
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