Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations

TL;DR

This paper establishes integral representations for nonnegative solutions of parabolic equations on noncompact Riemannian domains under semismall perturbation conditions, linking these solutions to boundary integrals and heat kernel properties.

Contribution

It introduces a new integral representation theorem for nonnegative solutions of parabolic equations on noncompact manifolds, connecting semismall perturbations with semi-intrinsic ultracontractivity.

Findings

Unique integral representation for solutions in finite time intervals

Representation involving Martin boundary for solutions on noncompact domains

Semismall perturbation implies semi-intrinsic ultracontractivity

Abstract

We consider nonnegative solutions of a parabolic equation in a cylinder $D \timesI$, where $D$ is a noncompact domain of a Riemannian manifold and $I =(0,T)$ with $0 < T \le \infty$ or $I=(-\infty,0)$. Under the assumption [SSP] (i.e., the constant function 1 is a semismall perturbation of the associated elliptic operator on $D$), we establish an integral representation theorem of nonnegative solutions: In the case $I =(0,T)$, any nonnegative solution is represented uniquely by an integral on $(D \times \{0 \}) \cup (\partial_M D \times [0,T))$, where $\partial_M D$ is the Martin boundary of $D$ for the elliptic operator; and in the case $I=(-\infty,0)$, any nonnegative solution is represented uniquely by the sum of an integral on $\partial_M D \times (-\infty,0)$ and a constant multiple of a particular solution. We also show that [SSP] implies the condition [SIU] (i.e., the associated heat kernel is semi-intrinsically ultracontractive).