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On the equation ut = ∆uα + uβ

Published online by Cambridge University Press:  14 November 2011

Yuan-Wei Qi
Yuan-Wei Qi
Affiliation:
Mathematical Institute, Oxford University, Oxford, OX1 3LB, U.K Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A
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Synopsis

The Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 2 , 1993 , pp. 373 - 390
Copyright
Copyright © Royal Society of Edinburgh 1993

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