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Blow-up rates for a fractional heat equation


Authors: R. Ferreira and A. de Pablo
Journal: Proc. Amer. Math. Soc. 149 (2021), 2011-2018
MSC (2020): Primary 35B44, 35K57, 35R11
DOI: https://doi.org/10.1090/proc/15165
Published electronically: March 2, 2021
MathSciNet review: 4232193
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Abstract: We study the speed at which nonglobal solutions to the fractional heat equation \begin{equation*} u_t+(-\Delta )^{\alpha /2} u=u^p, \end{equation*} with $0<\alpha <2$ and $p>1$, tend to infinity. We prove that, assuming either $p<p_F\equiv 1+\alpha /N$ or $u$ is strictly increasing in time, then for $t$ close to the blow-up time $T$ it holds that $\|u(\cdot ,t)\|_\infty \sim (T-t)^{-\frac 1{p-1}}$. The proofs use elementary tools, such as rescaling or comparison arguments.


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Additional Information

R. Ferreira
Affiliation: Departamento de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: raul_ferreira@mat.ucm.es

A. de Pablo
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Spain
MR Author ID: 271374
Email: arturop@math.uc3m.es

Keywords: Fractional laplacian, blow-up, blow-up rates
Received by editor(s): October 31, 2019
Received by editor(s) in revised form: April 15, 2020
Published electronically: March 2, 2021
Additional Notes: Work supported by the Spanish project MTM2017-87596.
The first author was also supported by Grupo de Investigación UCM 920894.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2021 American Mathematical Society