Advances in Differential Equations

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

Ugur G. Abdulla

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 18 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Abdulla, Ugur G. Wiener's criterion at $\infty$ for the heat equation. Adv. Differential Equations 13 (2008), no. 5-6, 457--488.

Export citation