English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

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 associatedelliptic operator on $D$), we establish an integral representation theorem ofnonnegative solutions: In the case $I =(0,T)$, any nonnegative solution isrepresented 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 theelliptic operator; and in the case $I=(-\infty,0)$, any nonnegative solution isrepresented uniquely by the sum of an integral on $\partial_M D \times(-\infty,0)$ and a constant multiple of a particular solution. We also showthat [SSP] implies the condition [SIU] (i.e., the associated heat kernel issemi-intrinsically ultracontractive).

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