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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:265

Article

Ghoul, T.¹  Ibrahim, S.^1,2  Nguyen, V. T.¹ 

[1] New York Univ Abu Dhabi, Dept Math, POB 129188, Abu Dhabi, U Arab Emirates

[2] Univ Victoria, Dept Math & Stat, POB 3060 STN CSC, Victoria, BC V8P 5C3, Canada

关键词: Wave maps;    Blowup solution;    Blowup profile;    Stability;   

DOI  :  10.1016/j.jde.2018.04.058

来源: Elsevier

【 摘 要 】

We consider the energy supercritical wave maps from R-d into the d-sphere S-d with d >= 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation partial derivative(2)(t)u = partial derivative(2)(r)u + (d - 1)/r partial derivative(r)u - (d - 1)/2r(2) 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) similar to Q(r/lambda(t)), where Q is the stationary solution of the equation and the speed is given by the quantized rates lambda(t) similar to c(u) (T - t)(l/gamma), l is an element of N*, l > gamma = gamma(d) is an element of (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, Raphael and Rodnianski [49] for the energy supercritical nonlinear Schrodinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem. Crown Copyright (C) 2018 Published by Elsevier Inc. All rights reserved.

【 授权许可】

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