Advances in Differential Equations

Wiener's criterion at $\infty$ for the heat equation

Ugur G. Abdulla

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Abstract

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 $0 < \epsilon < 1$.

Article information

Source
Adv. Differential Equations, Volume 13, Number 5-6 (2008), 457-488.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867342

Mathematical Reviews number (MathSciNet)
MR2482395

Zentralblatt MATH identifier
1160.35425

Subjects
Primary: 35K05: Heat equation
Secondary: 60J65: Brownian motion [See also 58J65]

Citation

Abdulla, Ugur G. Wiener's criterion at $\infty$ for the heat equation. Adv. Differential Equations 13 (2008), no. 5-6, 457--488. https://projecteuclid.org/euclid.ade/1355867342


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