Vol. 14, No. 5, 2021

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Existence and stability of infinite time bubble towers in the energy critical heat equation

Manuel del Pino, Monica Musso and Juncheng Wei

Vol. 14 (2021), No. 5, 1557–1598
Abstract

We consider the energy-critical heat equation in n for n 6

ut = Δu + |u| 4 n2 u in n × (0,), u( ,0) = u0 in n,

which corresponds to the L2-gradient flow of the Sobolev-critical energy

J(u) =ne[u],e[u] := 1 2|u|2 n 2 2n |u| 2n n2 .

Given any k 2 we find an initial condition u0 that leads to sign-changing solutions with multiple blow-up at a single point (tower of bubbles) as t +. It has the form of a superposition with alternate signs of singularly scaled Aubin–Talenti solitons,

u(x,t) = j=1k(1)j1μ jn2 2 U( x μj) + o(1) as t +,

where U(y) is the standard soliton

U(y) = αn( 1 1 + |y|2)n2 2

and

μj(t) = βjtαj ,αj = 1 2((n 2 n 6)j1 1)

if n 7. For n = 6, the rate of the μj(t) is different and it is also discussed. Letting δ0 be the Dirac mass, we have energy concentration of the form

e[u( ,t)] e[U] (k 1)Snδ0 as t +,

where Sn = J(U). The initial condition can be chosen radial and compactly supported. We establish the codimension k + n(k 1) stability of this phenomenon for perturbations of the initial condition that have space decay u0(x) = O(|x|α), α > (n 2)2, which yields finite energy of the solution.

Keywords
energy critical heat equation, infinite time blow-up
Mathematical Subject Classification 2010
Primary: 35K58
Secondary: 35B44
Milestones
Received: 2 October 2019
Revised: 3 December 2019
Accepted: 5 February 2020
Published: 22 August 2021
Authors
Manuel del Pino
Department of Mathematical Sciences
University of Bath
Bath
United Kingdom
Departamento de Ingeniería Matemática - CMM
Universidad de Chile
Santiago
Chile
Monica Musso
Department of Mathematical Sciences
University of Bath
Bath
United Kingdom
Juncheng Wei
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada