On the equation ut = ∆uα + uβ
1993 ◽
Vol 123
(2)
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pp. 373-390
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Keyword(s):
Blow Up
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SynopsisThe 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.
1990 ◽
Vol 50
(1)
◽
pp. 108-124
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2003 ◽
Vol 05
(03)
◽
pp. 369-400
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1991 ◽
Vol 38
(1)
◽
pp. 33-53
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1997 ◽
Vol 07
(04)
◽
pp. 957-962
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1982 ◽
Vol 4
(3)
◽
pp. 366-393
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2005 ◽
Vol 135
(3)
◽
pp. 585-602