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Asymptotics of the fast diffusion equation via entropy estimates
- Publication Year :
- 2007
-
Abstract
- We consider non-negative solutions of the fast diffusion equation $u_t=\Delta u^m$ with $m \in (0,1)$, in the Euclidean space R^d, d?3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to $t\to\infty$ for $m\ge m_c=(d-2)/d$, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if $m\ge m_c$, or close enough to the extinction time if m < mc. Such results are new in the range $m\le m_c$ where previous approaches fail. In the range mc < m < 1 we improve on known results.
- Subjects :
- Mathematics - Analysis of PDEs
35B40
35K55
39B62
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.0704.2372
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s00205-008-0155-z