Abstract
AbstractIn this expository note, we prove a localized bubbling result for solutions of the energy critical nonlinear heat equation with bounded$$\dot{H} ^1$$H˙1norm. The proof uses a combination of Gérard’s profile decomposition (ESAIM Control Optim. Calc. Var.3: 213–233, 1998), concentration compactness techniques in the spirit of Duyckaerts, Kenig, and Merle’s seminal work (Geom. Funct. Anal.22: 639–698, 2012), and a virial argument in the spirit of Jia and Kenig’s work (Amer. J. Math.139: 1521–1603, 2017) to deduce the vanishing of the error in the neck regions between the bubbles. The argument is based closely on an analogous lemma proved in the author’s recent work with Jendrej (arXiv:2210.14963, 2022) on the equivariant harmonic map heat flow in dimension two.
Funder
NSF
Alfred P. Sloan Foundation
Publisher
Springer Science and Business Media LLC
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