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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 
Keywords
 energy critical heat equation
 infinite time blowup
ASJC Scopus subject areas
 Analysis
 Numerical Analysis
 Applied Mathematics
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Concentration phenomena in nonlinear analysis
Engineering and Physical Sciences Research Council
27/04/20 → 26/04/23
Project: Research council