抛物型方程非负解的半小扰动、半内禀超压缩性和整表示
Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations
摘要：
考虑圆柱$D \timesI$中抛物型方程的非负解，其中$D$是黎曼流形的非紧区域，$I =(0,T)$具有$0 < T \le \infty$或$I=(-\infty,0)$ .假设[SSP]（即，常数函数1是椭圆算子在$D$上的半小扰动），我们建立了非负解的积分表示定理：在$I =(0,T)$情形下，任何非负解唯一地表示为$(D \times \{0 \}) \cup (\partial_M D \times [0,T))$上的积分，其中$\partial_M D$是椭圆算子在$D$上的Martin边界;而在$I=(-\infty,0)$的情况下，任何非负解唯一地由$\partial_M D \times (-\infty,0)$上的积分与特定解的常数倍数之和表示。我们也证明了[SSP]隐含条件[SIU]（即，相关联的热核是半本征超收缩的）。
英文摘要：
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).
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