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

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).

35 pages