Global, decaying solutions of a focusing energy-critical heat equation in ℝ⁴

2017/07/24 by Stephen Gustafson, Gustafson, Stephen, Dimitrios Roxanas +1
Mathematics · #35K55 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35K55

paper · pdf · doi:10.48550/arxiv.1707.07644

41 pages

arxiv created 2017/07/24 · arxiv updated 2017/07/25

Abstract

We study solutions of the focusing energy-critical nonlinear heat equation u_t = Δu - |u|²u in ℝ⁴. We show that solutions emanating from initial data with energy and H¹-norm below those of the stationary solution W are global and decay to zero, via the "concentration-compactness plus rigidity" strategy of Kenig-Merle. First, such global solutions are shown to dissipate to zero, using a refinement of the small data theory and the L²-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza, Seregin and Sverak in an argument similar to that of Kenig and Koch for the Navier-Stokes equations.

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