Singular solutions for fractional parabolic boundary value problems

TL;DR

This paper develops a theory for solving fractional parabolic boundary value problems with singular boundary data, extending classical heat equation results to fractional and more general integro-differential operators, and analyzes spectral properties of the associated semigroup.

Contribution

It introduces a comprehensive existence and uniqueness framework for fractional parabolic problems with singular data, including a spectral theory extension for various fractional operators.

Findings

Established existence and uniqueness of solutions with singular boundary data.

Extended spectral Weyl law to censored fractional Laplacian.

Derived bounds on fractional heat kernels.

Abstract

The standard problem for the classical heat equation posed in a bounded domain $\Omega$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution $u(t,x)$ blows up as $x$ approaches $\partial \Omega$ in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel.