TY - JOUR
T1 - Type II finite time blow-up for the energy critical heat equation in R4
AU - del Pino, Manuel
AU - Musso, Monica
AU - Wei, Juncheng
AU - Zhou, Yifu
N1 - 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.
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.
KW - Construction of blow-up solution
KW - Energy critical heat equation
KW - Inner–outer gluing method
KW - Nondegeneracy
KW - Type II finite time blow-up
UR - http://www.scopus.com/inward/record.url?scp=85082526284&partnerID=8YFLogxK
U2 - 10.3934/dcds.2020052
DO - 10.3934/dcds.2020052
M3 - Article
AN - SCOPUS:85082526284
SN - 1078-0947
VL - 40
SP - 3327
EP - 3335
JO - Discrete and Continuous Dynamical Systems- Series A
JF - Discrete and Continuous Dynamical Systems- Series A
IS - 6
ER -