Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case

详细信息    查看全文

• 作者：C. Cortá ; zar^a ; ^{ccortaza@mat.puc.cl" class="auth_mail" title="E-mail the corresponding author} ; M. Elgueta^a ; ^{melgueta@mat.puc.cl" class="auth_mail" title="E-mail the corresponding author} ; F. Quiró ; s^b ; ^{fernando.quiros@uam.es" class="auth_mail" title="E-mail the corresponding author} ; N. Wolanski^c ; ^d ; ^{wolanski@dm.uba.ar" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlocal diffusion ; Exterior domain ; Asymptotic behavior ; Matched asymptotics
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：436
• 期：1
• 页码：586-610
• 全文大小：520 K

文摘

We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, the MathML source">∂_tu=J⁎u−u${\partial }_{t}u=J⁎u-u$, where J   is a smooth, radially symmetric kernel with support the MathML source">B_d(0)⊂R²$B_d(0)\subset {\mathbb{R}}^2$. The problem is set in an exterior two-dimensional domain which excludes a hole the MathML source">H$\mathcal{H}$, and with zero Dirichlet data on the MathML source">H$\mathcal{H}$. In the far field scale, the MathML source">ξ₁≤|x|t^−1/2≤ξ₂${\xi }_1\le |x|t^{-1/2}\le {\xi }_2$ with the MathML source">ξ₁,ξ₂>0${\xi }_1,{\xi }_2>0$, the scaled function the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si6.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=915f026cd8131d46aeb22556d587f4ec">the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X15011270-si6.gif">$\log tu(x,t)$ behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by J  . The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic ‘logarithmic momentum' of the solution, the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si7.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=02ec2fc2b18ccdc8aa82f444ae6ce50b">the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X15011270-si7.gif">${\lim }_{t\to \infty }{\int }_{{\mathbb{R}}^2}u(x,t)\log |x|dx$. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, the MathML source">|x|≤t^1/2h(t)$|x|\le t^{1/2}h(t)$ with the MathML source">lim_t→∞⁡h(t)=0${\lim }_{t\to \infty }h(t)=0$, the scaled function the MathML source">t(log⁡t)²u(x,t)/log⁡|x|$t{(\log t)}^{2}u(x,t)/\log |x|$ converges to a multiple of the MathML source">ϕ(x)/log⁡|x|$\varphi (x)/\log |x|$, where ϕ   is the unique stationary solution of the problem that behaves as the MathML source">log⁡|x|$\log |x|$ when the MathML source">|x|→∞$|x|\to \infty$. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, the MathML source">|x|≥t^1/2g(t)$|x|\ge t^{1/2}g(t)$ with ce458527b4a8e5d" title="Click to view the MathML source">g(t)→∞$g(t)\to \infty$, the solution is proved to be of order the MathML source">o((tlog⁡t)⁻¹)$o({(t\log t)}^{-1})$.