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2008 Wiener's criterion at $\infty$ for the heat equation
Ugur G. Abdulla
Adv. Differential Equations 13(5-6): 457-488 (2008). DOI: 10.57262/ade/1355867342


This paper introduces a notion of regularity (or irregularity) of the point at infinity ($\infty$) for the unbounded open set $\Omega\subset \mathbb R ^{N+1}$ concerning the heat equation, according as whether the parabolic measure of $\infty$ is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the parabolic Dirichlet problem in arbitrary unbounded open subset of $ \mathbb R ^{N+1}$ is established. It is expressed in terms of the Wiener's criterion for the regularity of $\infty$. A geometric iterated logarithm test for the well-posedness of the parabolic Dirichlet problem in arbitrary open subset of $ \mathbb R ^{N+1}$ ($N \geq 2$) is proved. A domain is produced for which the parabolic Dirichlet problem always has a unique bounded solution for the heat equation $u_t=\Delta u$, and infinitely many for the equation $u_t=(1-\epsilon)\Delta u$ for all $0 < \epsilon < 1$.


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Ugur G. Abdulla. "Wiener's criterion at $\infty$ for the heat equation." Adv. Differential Equations 13 (5-6) 457 - 488, 2008.


Published: 2008
First available in Project Euclid: 18 December 2012

zbMATH: 1160.35425
MathSciNet: MR2482395
Digital Object Identifier: 10.57262/ade/1355867342

Primary: 35K05
Secondary: 60J65

Rights: Copyright © 2008 Khayyam Publishing, Inc.


Vol.13 • No. 5-6 • 2008
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