Type II finite time blow-up for the energy critical heat equation in R4

Manuel del Pino; Monica Musso; Juncheng Wei; Yifu Zhou

doi:10.3934/dcds.2020052

Type II finite time blow-up for the energy critical heat equation in R⁴

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou

Department of Mathematical Sciences

University of British Columbia

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

9 Citations (SciVal)

Abstract

We consider the Cauchy problem for the energy critical heat equation ut = ∆u + u³ in R⁴ × (0, T), (1) ⁽u(·, 0) = u₀ in R⁴. We find that for given points q1, q2, . . ., q_k and any sufficiently small T > 0 there is an initial condition u₀ such that the solution u(x, t) of (1) blows up at exactly those k points with a type II rate, namely larger than (T − t)⁻ 2 . In 1 fact ku(·, t)k∞ ∼ (T − t)⁻ ¹ log²(T − t). 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)	3327-3335
Number of pages	9
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	40
Issue number	6
DOIs	https://doi.org/10.3934/dcds.2020052
Publication status	Published - 30 Jun 2020

Bibliographical note

Funding Information:
2010 Mathematics Subject Classification. Primary: 35K58; Secondary: 35B40. Key words and phrases. Energy critical heat equation, type II finite time blow-up, inner–outer gluing method, construction of blow-up solution, nondegeneracy. The first author has been supported by a UK Royal Society Research Professorship and Grant PAI AFB-170001, Chile. The second author has been partly supported by Fondecyt grant 1160135, Chile. The research of the third author is partially supported by NSERC of Canada. ∗ Corresponding author: Manuel del Pino.

Funding Information:
Acknowledgments. We thank the anonymous referee for a thorough review and several useful comments. M. del Pino has been supported by a UK Royal Society Research Professorship and Grant PAI AFB-170001, Chile. M. Musso has been partly supported by Fondecyt grant 1160135, Chile. The research of J. Wei is partially supported by NSERC of Canada.

Publisher Copyright:
© 2020 American Institute of Mathematical Sciences. All rights reserved.

Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

Keywords

Construction of blow-up solution
Energy critical heat equation
Inner–outer gluing method
Nondegeneracy
Type II finite time blow-up

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.3934/dcds.2020052Licence: CC BY-NC-ND

Cite this

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title = "Type II finite time blow-up for the energy critical heat equation in R4",

abstract = "We consider the Cauchy problem for the energy critical heat equation ut = ∆u + u3 in R4 × (0, T), (1) (u(·, 0) = u0 in R4. We find that for given points q1, q2, . . ., qk and any sufficiently small T > 0 there is an initial condition u0 such that the solution u(x, t) of (1) blows up at exactly those k points with a type II rate, namely larger than (T − t)− 2 . In 1 fact ku(·, t)k∞ ∼ (T − t)− 1 log2(T − t). The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.",

keywords = "Construction of blow-up solution, Energy critical heat equation, Inner–outer gluing method, Nondegeneracy, Type II finite time blow-up",

author = "{del Pino}, Manuel and Monica Musso and Juncheng Wei and Yifu Zhou",

note = "Funding Information: 2010 Mathematics Subject Classification. Primary: 35K58; Secondary: 35B40. Key words and phrases. Energy critical heat equation, type II finite time blow-up, inner–outer gluing method, construction of blow-up solution, nondegeneracy. The first author has been supported by a UK Royal Society Research Professorship and Grant PAI AFB-170001, Chile. The second author has been partly supported by Fondecyt grant 1160135, Chile. The research of the third author is partially supported by NSERC of Canada. ∗ Corresponding author: Manuel del Pino. Funding Information: Acknowledgments. We thank the anonymous referee for a thorough review and several useful comments. M. del Pino has been supported by a UK Royal Society Research Professorship and Grant PAI AFB-170001, Chile. M. Musso has been partly supported by Fondecyt grant 1160135, Chile. The research of J. Wei is partially supported by NSERC of Canada. Publisher Copyright: {\textcopyright} 2020 American Institute of Mathematical Sciences. All rights reserved. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.",

year = "2020",

month = jun,

day = "30",

doi = "10.3934/dcds.2020052",

language = "English",

volume = "40",

pages = "3327--3335",

journal = "Discrete and Continuous Dynamical Systems- Series A",

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AU - del Pino, Manuel

AU - Musso, Monica

AU - Wei, Juncheng

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PY - 2020/6/30

Y1 - 2020/6/30

N2 - We consider the Cauchy problem for the energy critical heat equation ut = ∆u + u3 in R4 × (0, T), (1) (u(·, 0) = u0 in R4. We find that for given points q1, q2, . . ., qk and any sufficiently small T > 0 there is an initial condition u0 such that the solution u(x, t) of (1) blows up at exactly those k points with a type II rate, namely larger than (T − t)− 2 . In 1 fact ku(·, t)k∞ ∼ (T − t)− 1 log2(T − t). 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 = ∆u + u3 in R4 × (0, T), (1) (u(·, 0) = u0 in R4. We find that for given points q1, q2, . . ., qk and any sufficiently small T > 0 there is an initial condition u0 such that the solution u(x, t) of (1) blows up at exactly those k points with a type II rate, namely larger than (T − t)− 2 . In 1 fact ku(·, t)k∞ ∼ (T − t)− 1 log2(T − t). The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.

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Type II finite time blow-up for the energy critical heat equation in R⁴

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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Concentration phenomena in nonlinear analysis

Cite this

Type II finite time blow-up for the energy critical heat equation in R4

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Concentration phenomena in nonlinear analysis

Cite this

Type II finite time blow-up for the energy critical heat equation in R⁴