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Abstract
We consider the energycritical heat equation in ℝ^{n} for n ≥ 6 (Formula presented) which corresponds to the L^{2}gradient flow of the Sobolevcritical energy (Formula presented) Given any k ≥ 2 we find an initial condition u_{0} that leads to signchanging solutions with multiple blowup at a single point (tower of bubbles) as t →+∞. It has the form of a superposition with alternate signs of singularly scaled AubinTalenti solitons, (Formula presented) where U(y) is the standard soliton (Formula presented) and (Formula presented) if n ≥ 7. For n = 6, the rate of the μ_{j}(t) is different and it is also discussed. Letting δ_{0} be the Dirac mass, we have energy concentration of the form (Formula presented) where S_{n} = J(U). The initial condition can be chosen radial and compactly supported. We establish the codimension k+n (k  l) stability of this phenomenon for perturbations of the initial condition that have space decay u_{0}(x) = O(x^{−α}), α > (n  2)/2, which yields finite energy of the solution.
Original language  English 

Pages (fromto)  15571598 
Number of pages  42 
Journal  Analysis and PDE 
Volume  14 
Issue number  5 
DOIs  
Publication status  Published  22 Aug 2021 
Bibliographical note
Funding Information:Del Pino has been supported by a UK Royal Society Research Professorship and Grant PAI AFB170001, Chile. Musso has been partly supported by Fondecyt Grant 1160135, Chile. The research of Wei is partially supported by the NSERC of Canada.
Keywords
 energy critical heat equation
 infinite time blowup
ASJC Scopus subject areas
 Analysis
 Numerical Analysis
 Applied Mathematics
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 1 Finished

Concentration phenomena in nonlinear analysis
Engineering and Physical Sciences Research Council
27/04/20 → 31/03/24
Project: Research council