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Approximation theorems for parabolic equations and movement of local hot spots | | Alberto Enciso
; M. Ángeles García-Ferrero
; Daniel Peralta-Salas
; | | Date: |
10 Oct 2017 | | Abstract: | We prove a global approximation theorem for a general parabolic operator $L$,
which asserts that if $v$ satisfies the equation $Lv=0$ in a spacetime region
$Omegasubsetmathbf{R}^{n+1}$ satisfying certain necessary topological
condition, then it can be approximated in a H"older norm by a global solution
$u$ to the equation. If $Omega$ is compact and the operator $L$ satisfies
certain technical conditions (e.g., when it is the usual heat equation), the
global solution $u$ can be shown to fall off in space and time. This result is
next applied 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 discussed too. | | Source: | arXiv, 1710.3782 | | Services: | Forum | Review | PDF | Favorites | |
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