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1 April 2019 Approximation theorems for parabolic equations and movement of local hot spots
Alberto Enciso, MªÁngeles García-Ferrero, Daniel Peralta-Salas
Duke Math. J. 168(5): 897-939 (1 April 2019). DOI: 10.1215/00127094-2018-0058


We prove a global approximation theorem for a general parabolic operator L, which asserts that if v satisfies the equation Lv=0 in a space-time region ΩRn+1 satisfying a certain necessary topological condition, then it can be approximated in a Hölder norm by a global solution u to the equation. If Ω is compact and L is the usual heat operator, then one can instead approximate the local solution v by the unique solution that falls off at infinity to the Cauchy problem with a suitably chosen smooth, compactly supported initial datum. We then apply these results to prove the existence of global solutions to the equation Lu=0 with a local hot spot that moves along a prescribed curve for all time, up to a uniformly small error. Global solutions that exhibit isothermic hypersurfaces of prescribed topologies for all times and applications to the heat equation on the flat torus are also discussed.


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Alberto Enciso. MªÁngeles García-Ferrero. Daniel Peralta-Salas. "Approximation theorems for parabolic equations and movement of local hot spots." Duke Math. J. 168 (5) 897 - 939, 1 April 2019.


Received: 11 October 2017; Revised: 23 October 2018; Published: 1 April 2019
First available in Project Euclid: 2 March 2019

zbMATH: 07055196
MathSciNet: MR3934592
Digital Object Identifier: 10.1215/00127094-2018-0058

Primary: 35B05
Secondary: 35K10

Rights: Copyright © 2019 Duke University Press


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Vol.168 • No. 5 • 1 April 2019
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