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arXiv 1004.0717math.AP
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具吸收项和有界初值的非局部扩散方程的长时间行为

Large Time Behavior of a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data

Joana Terra, Noemi Wolanski

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摘要：

研究了Cauchy问题$u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$，$u(x,0)=u_0(x)\in L^\infty$的非负解的大时间行为，其中$|x|^αu_0(x)\to A>0$为$|x|\to\infty$ .我们的主要目标之一是研究关键情况下$p=1+2/α$的$0<α<N$，在以前的文章中，我们证明了$t^{α/2}|u(x,t)-U(x,t)|\to 0$，其中$U$是解决方案的热方程与初始数据$U(x,0)=C_{A,N}|x|^{-α}$的吸收。我们的证明，涉及序列的重新缩放的解决方案，使我们能够建立大的时间行为的解决方案具有更一般的nonintegrable初始数据$u_0$在超临界的情况下，也在临界情况下（$p=1+2/N$）的有界和可积的$u_0$。

英文摘要：

We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^αu_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals is the study of the critical case $p=1+2/α$ for $0<α<N$, left open in previous articles, for which we prove that $t^{α/2}|u(x,t)-U(x,t)|\to 0$ where $U$ is the solution of the heat equation with absorption with initial datum $U(x,0)=C_{A,N}|x|^{-α}$. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data $u_0$ in the supercritical case and also in the critical case ($p=1+2/N$) for bounded and integrable $u_0$.

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