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arXiv 1302.1786math.AP
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具有非局部扩散的heat equation的Widder型定理

A Widder's type Theorem for the heat equation with nonlocal diffusion

Begoña Barrios, Ireneo Peral, Fernando Soria, Enrico Valdinoci

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摘要：

本文的主要目的是证明问题$$ u_t+(-Δ)^{α/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<α<2, $$的每一个非负{ \it强解} $u(x,t)$都可以写成$$u(x,t)=\int_{\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\, dy},$$，其中$$P_{t}(x)=\frac{1}{t^{n/α}}P\left(\frac{x}{t^{1/α}}\right), $$和$$ P(x):=\int_{\mathbb{R}^{n}}{e^{ix\cdotξ-|ξ|^α}dξ}. $$。V. Widder在\cite{W0}的非局部扩散框架。

英文摘要：

The main goal of this work is to prove that every non-negative {\it strong solution} $u(x,t)$ to the problem $$ u_t+(-Δ)^{α/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<α<2, $$ can be written as $$u(x,t)=\int_{\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\, dy},$$ where $$P_{t}(x)=\frac{1}{t^{n/α}}P\left(\frac{x}{t^{1/α}}\right), $$ and $$ P(x):=\int_{\mathbb{R}^{n}}{e^{ix\cdotξ-|ξ|^α}dξ}. $$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by D. V. Widder in \cite{W0} to the nonlocal diffusion framework.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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