Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data
We study the large time behavior of nonnegative solutions of the Cauchy problem u t = R J ( x − y )( u ( y,t ) − u ( x,t )) dy − u p , u ( x, 0) = u 0 ( x ) ∈ L ∞ , where | x | α u 0 ( x ) → A > 0 as | x |→∞ . One of our main goals is the study of the critical case p = 1 + 2 /α for 0 < α <...
| Authors: | , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 2011 |
| Country: | Argentina |
| Institution: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repository: | CONICET Digital (CONICET) |
| Language: | English |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/14924 |
| Online Access: | http://hdl.handle.net/11336/14924 |
| Access Level: | Open access |
| Keyword: | Nonlocal diffusion Large time behavior https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Summary: | We study the large time behavior of nonnegative solutions of the Cauchy problem u t = R J ( x − y )( u ( y,t ) − u ( x,t )) dy − u p , u ( x, 0) = u 0 ( x ) ∈ L ∞ , where | x | α u 0 ( x ) → A > 0 as | x |→∞ . 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 ) | → 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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