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

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• 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, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si1.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=14998c583b62f787d4dad68d8c56495a" title="Click to view the MathML source">∂_tu=J⁎u−uclass="mathContainer hidden">class="mathCode">${\partial }_{t}u=J⁎u-u$, where J   is a smooth, radially symmetric kernel with support class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si2.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=05ca97278e34c77cef79c1ac8924fdde" title="Click to view the MathML source">B_d(0)⊂R²class="mathContainer hidden">class="mathCode">$B_d(0)\subset {\mathbb{R}}^2$. The problem is set in an exterior two-dimensional domain which excludes a hole class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si214.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=79c970dd49f7a920725ea8adfd4b1581" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$\mathcal{H}$, and with zero Dirichlet data on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si214.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=79c970dd49f7a920725ea8adfd4b1581" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$\mathcal{H}$. In the far field scale, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si4.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=34ec1db22fc0800a6b76078ac96d7549" title="Click to view the MathML source">ξ₁≤|x|t^−1/2≤ξ₂class="mathContainer hidden">class="mathCode">${\xi }_1\le |x|t^{-1/2}\le {\xi }_2$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si5.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=f2199a5fa29276c9bb2cba499fdcb912" title="Click to view the MathML source">ξ₁,ξ₂>0class="mathContainer hidden">class="mathCode">${\xi }_1,{\xi }_2>0$, the scaled function class="mathmlsrc">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">class="imgLazyJSB inlineImage" height="16" width="77" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X15011270-si6.gif">class="mathContainer hidden">class="mathCode">$\log tu(x,t)$ 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, class="mathmlsrc">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">class="imgLazyJSB inlineImage" height="19" width="192" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X15011270-si7.gif">class="mathContainer hidden">class="mathCode">${\lim }_{t\to \infty }{\int }_{{\mathbb{R}}^2}u(x,t)\log |x|dx$. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si8.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=aab4b5a96727c1b1513f52853ea7eee4" title="Click to view the MathML source">|x|≤t^1/2h(t)class="mathContainer hidden">class="mathCode">$|x|\le t^{1/2}h(t)$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si9.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=0a808c68f5296b2b2ea571fae50ae287" title="Click to view the MathML source">lim_t→∞⁡h(t)=0class="mathContainer hidden">class="mathCode">${\lim }_{t\to \infty }h(t)=0$, the scaled function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si10.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=1e543d469fad35223a153f3e92888d71" title="Click to view the MathML source">t(log⁡t)²u(x,t)/log⁡|x|class="mathContainer hidden">class="mathCode">$t{(\log t)}^{2}u(x,t)/\log |x|$ converges to a multiple of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si11.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=0affa1d9497a788f9a053198b4cf8c6a" title="Click to view the MathML source">ϕ(x)/log⁡|x|class="mathContainer hidden">class="mathCode">$\varphi (x)/\log |x|$, where ϕ   is the unique stationary solution of the problem that behaves as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si12.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=c446743ec4c836032970468fe14592f9" title="Click to view the MathML source">log⁡|x|class="mathContainer hidden">class="mathCode">$\log |x|$ when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si13.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=bc76169a294fde606f825ffa5e0b3f22" title="Click to view the MathML source">|x|→∞class="mathContainer hidden">class="mathCode">$|x|\to \infty$. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si14.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=3ed0bfa696617f2e442ea520973a721b" title="Click to view the MathML source">|x|≥t^1/2g(t)class="mathContainer hidden">class="mathCode">$|x|\ge t^{1/2}g(t)$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si15.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=11a7ad5b27814fd2bce458527b4a8e5d" title="Click to view the MathML source">g(t)→∞class="mathContainer hidden">class="mathCode">$g(t)\to \infty$, the solution is proved to be of order class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15011270&_mathId=si16.gif&_user=111111111&_pii=S0022247X15011270&_rdoc=1&_issn=0022247X&md5=809151339382a0fc7a9b6682d91072ce" title="Click to view the MathML source">o((tlog⁡t)⁻¹)class="mathContainer hidden">class="mathCode">$o({(t\log t)}^{-1})$.