Type II Blow-up in the 5-dimensional Energy Critical Heat Equation

Manuel del Pino; Monica Musso; Jun Cheng Wei

doi:10.1007/s10114-019-8341-5

Type II Blow-up in the 5-dimensional Energy Critical Heat Equation

Manuel del Pino, Monica Musso, Jun Cheng Wei

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

27 Citations (SciVal)

Abstract

We consider the Cauchy problem for the energy critical heat equation {ut=Δu+|u|4n−2uinRn×(0,T)u(⋅,0)=u0inRn in dimension n = 5. More precisely we find that for given points q ₁ ,q ₂ ,..,q _k and any sufficiently small T > 0 there is an initial condition u ₀ such that the solution u(x,t) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size ~(T-t) ^-α for α > 34. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.

Original language	English
Pages (from-to)	1027-1042
Number of pages	16
Journal	Acta Mathematica Sinica, English Series
Volume	35
Issue number	6
Early online date	20 May 2019
DOIs	https://doi.org/10.1007/s10114-019-8341-5
Publication status	Published - 1 Jun 2019

Funding

Received August 20, 2018, accepted December 21, 2018 M. del Pino has been supported by a UK Royal Society Research Professorship and Fondo Basal CMM-Chile. M. Musso has been partly supported by grants Fondecyt 1160135, Chile. The research of J. Wei is partially supported by NSERC of Canada

Keywords

35B40
35K58
bubbling phenomena
critical parabolic equations
Singularity formation

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.1007/s10114-019-8341-5

Cite this

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title = "Type II Blow-up in the 5-dimensional Energy Critical Heat Equation",

abstract = " We consider the Cauchy problem for the energy critical heat equation {ut=Δu+|u|4n−2uinRn×(0,T)u(⋅,0)=u0inRn in dimension n = 5. More precisely we find that for given points q 1 ,q 2 ,..,q k and any sufficiently small T > 0 there is an initial condition u 0 such that the solution u(x,t) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size ~(T-t) -α for α > 34. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles. ",

keywords = "35B40, 35K58, bubbling phenomena, critical parabolic equations, Singularity formation",

author = "{del Pino}, Manuel and Monica Musso and Wei, {Jun Cheng}",

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AU - del Pino, Manuel

AU - Musso, Monica

AU - Wei, Jun Cheng

PY - 2019/6/1

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N2 - We consider the Cauchy problem for the energy critical heat equation {ut=Δu+|u|4n−2uinRn×(0,T)u(⋅,0)=u0inRn in dimension n = 5. More precisely we find that for given points q 1 ,q 2 ,..,q k and any sufficiently small T > 0 there is an initial condition u 0 such that the solution u(x,t) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size ~(T-t) -α for α > 34. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.

AB - We consider the Cauchy problem for the energy critical heat equation {ut=Δu+|u|4n−2uinRn×(0,T)u(⋅,0)=u0inRn in dimension n = 5. More precisely we find that for given points q 1 ,q 2 ,..,q k and any sufficiently small T > 0 there is an initial condition u 0 such that the solution u(x,t) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size ~(T-t) -α for α > 34. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.

KW - 35B40

KW - 35K58

KW - bubbling phenomena

KW - critical parabolic equations

KW - Singularity formation

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JO - Acta Mathematica Sinica, English Series

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Type II Blow-up in the 5-dimensional Energy Critical Heat Equation

Abstract

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Keywords

ASJC Scopus subject areas

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Concentration phenomena in nonlinear analysis

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Type II Blow-up in the 5-dimensional Energy Critical Heat Equation

Abstract

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

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Concentration phenomena in nonlinear analysis

Cite this