Archiv der Mathematik

, 97:353 | Cite as

Extinction for a fast diffusion equation with a nonlinear nonlocal source



In this article, the authors establish conditions for the extinction of solutions, in finite time, of the fast diffusion equation \({u_t=\Delta u^m+a\int_\Omega u^p(y,t)\,{d}y,\ 0 < m < 1,}\) in a bounded domain \({\Omega\subset R^N}\) with N > 2. More precisely speaking, it is shown that if p > m, any solution with small initial data vanishes in finite time, and if p < m, the maximal solution is positive in Ω for all t > 0. For the critical case p = m, whether the solutions vanish in finite time or not depends on the value of , where \({\mu=\int_{\Omega}\varphi(x)\,{d}x}\) and \({\varphi}\) is the unique positive solution of the elliptic problem \({-\Delta\varphi(x)=1,\ x\in \Omega; \varphi(x)=0,\ x\in\partial\Omega}\) .

Mathematics Subject Classification (2010)

Primary 35K55 Secondary 35K57 


Fast diffusion equation Nonlocal source Extinction in finite time 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of MathematicsJilin UniversityChangchunPeople’s Republic of China

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