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2021 Existence and stability of infinite time bubble towers in the energy critical heat equation
Manuel del Pino, Monica Musso, Juncheng Wei
Anal. PDE 14(5): 1557-1598 (2021). DOI: 10.2140/apde.2021.14.1557

Abstract

We consider the energy-critical heat equation in n for n6

ut=Δu+|u|4n2u 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]:=12|u|2n22n|u|2nn2.

Given any k2 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)j1μjn22U(xμj)+o(1) as t+,

where U(y) is the standard soliton

U(y)=αn(11+|y|2)n22

and

μj(t)=βjtαj,αj=12((n2n6)j11)

if n7. 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](k1)Snδ0 as t+,

where Sn=J(U). The initial condition can be chosen radial and compactly supported. We establish the codimension k+n(k1) stability of this phenomenon for perturbations of the initial condition that have space decay u0(x)=O(|x|α), α>(n2)2, which yields finite energy of the solution.

Citation

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Manuel del Pino. Monica Musso. Juncheng Wei. "Existence and stability of infinite time bubble towers in the energy critical heat equation." Anal. PDE 14 (5) 1557 - 1598, 2021. https://doi.org/10.2140/apde.2021.14.1557

Information

Received: 2 October 2019; Revised: 3 December 2019; Accepted: 5 February 2020; Published: 2021
First available in Project Euclid: 6 January 2022

Digital Object Identifier: 10.2140/apde.2021.14.1557

Subjects:
Primary: 35K58
Secondary: 35B44

Keywords: energy critical heat equation , infinite time blow-up

Rights: Copyright © 2021 Mathematical Sciences Publishers

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