Finite-time blowup for a supercritical defocusing nonlinear wave system

Abstract

We consider the global regularity problem for defocusing nonlinear wave systems u D .r‫ޒ‬ m F /.u/ on Minkowski spacetime ‫ޒ‬ 1Cd with d'Alembertian WD @ 2 t C P d iD1 @ 2 x i , where the field u W ‫ޒ‬ 1Cd !‫ޒ‬ m is vector-valued, and F W ‫ޒ‬ m !‫ޒ‬ is a smooth potential which is positive and homogeneous of order p C 1 outside of the unit ball for some p > 1.This generalises the scalar defocusing nonlinear wave (NLW) equation, in which m D 1 and F.v/ D 1=.p C 1/jvj pC1 .It is well known that in the energy-subcritical and energy-critical cases when d Ä 2 or d 3 and p Ä 1 C 4=.d 2/, one has global existence of smooth solutions from arbitrary smooth initial data u.0/; @ t u.0/, at least for dimensions d Ä 7. We study the supercritical case where d D 3 and p > 5. We show that in this case, there exists a smooth potential F for some sufficiently large m (in fact we can take m D 40), positive and homogeneous of order p C 1 outside of the unit ball, and a smooth choice of initial data u.0/; @ t u.0/ for which the solution develops a finite-time singularity.In fact the solution is discretely self-similar in a backwards light cone.The basic strategy is to first select the mass and energy densities of u, then u itself, and then finally design the potential F in order to solve the required equation.The Nash embedding theorem is used in the second step, explaining the need to take m relatively large.

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• Analysis & PDE - View
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF
• DataCite API - View
• Analysis & PDE - View
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF
• DataCite API - View