## Advances in Differential Equations

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

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

#### 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