Fractional p-Laplacian evolution equations

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• 作者：José ; M. Mazó ; n^a ; ^{mazon@uv.es" class="auth_mail" title="E-mail the corresponding author} ; Julio D. Rossi^b ; ^{jrossi@dm.uba.ar" class="auth_mail" title="E-mail the corresponding author} ; Juliá ; n Toledo^c ; ^{toledojj@uv.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：45G10 ; 45J05 ; 47H06
• 刊名：Journal de mathematiques pures et appliquees
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：105
• 期：6
• 页码：810-844
• 全文大小：552 K

文摘

In this paper we study the fractional p-Laplacian evolution equation given bymulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si2.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=64f0c3f8d802475f336f62be4503afbe" title="Click to view the MathML source">0<s<1$0<s<1$, mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si3.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=f04350c14ac9d8fdd5a040fc11234e69" title="Click to view the MathML source">p≥1$p\ge 1$. In a bounded domain Ω we deal with the Dirichlet problem by taking mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si4.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=96f60e0bf640aa064717943b03558c8f" title="Click to view the MathML source">A=R^N$A={\mathbb{R}}^N$ and mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si5.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=9ea824c6c0ddc085a443ea5f08c57553" title="Click to view the MathML source">u=0$u=0$ in mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si6.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=1cf7ca07b806341955811338e8ee6e78" title="Click to view the MathML source">R^N∖Ω${\mathbb{R}}^N\setminus \mathrm{\Omega }$, and the Neumann problem by taking mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si7.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=e44ff624e6336358f8722a9a4e5ee8f9" title="Click to view the MathML source">A=Ω$A=\mathrm{\Omega }$. We include here the limit case mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si8.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=c3aac8f5d6a6967b4bb92f6e97fa49e6" title="Click to view the MathML source">p=1$p=1$ that has the extra difficulty of giving a meaning to 2416000052&_mathId=si18.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=184a2613fd56463e2205e45160164a57">2416000052-si18.gif">$\frac{u(y)-u(x)}{|u(y)-u(x)|}$ when mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si10.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=bb1e8434752bd7146f42bece8128e94b" title="Click to view the MathML source">u(y)=u(x)$u(y)=u(x)$. We also consider the Cauchy problem in the whole mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si11.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=f01da517382f3574056c1ba10ad2bb6e" title="Click to view the MathML source">R^N${\mathbb{R}}^N$ by taking mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si12.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=b7c7a7e312cb28ac3981997d2d739557" title="Click to view the MathML source">A=Ω=R^N$A=\mathrm{\Omega }={\mathbb{R}}^N$. We find existence and uniqueness of strong solutions for each of the above mentioned problems. We also study the asymptotic behaviour of these solutions as mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si13.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=4211d8e6042fd199d796a45d57f216cb" title="Click to view the MathML source">t→∞$t\to \infty$. Finally, we recover the local p  -Laplacian evolution equation with Dirichlet or Neumann boundary conditions, and for the Cauchy problem, by taking the limit as mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782416000052&_mathId=si14.gif&_user=111111111&_pii=S0021782416000052&_rdoc=1&_issn=00217824&md5=8e349b44ac2382def3abc6efe7c17aea" title="Click to view the MathML source">s→1$s\to 1$ in the nonlocal problems multiplied by a suitable scaling constant.