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The critical exponents of parabolic equations and blow-up in Rn

Published online by Cambridge University Press:  14 November 2011

Yuan-wei Qi
Yuan-wei Qi
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong
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Extract

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 1 , 1998 , pp. 123 - 136
Copyright
Copyright © Royal Society of Edinburgh 1998

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