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Zbl 0892.35088
Qi, Yuan-wei
The critical exponents of parabolic equations and blow-up in $R^n$. (English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 128, No.1, 123-136 (1998). ISSN 0308-2105; ISSN 1473-7124/e

Summary: We study the Cauchy problem in $\bbfR^n$ of general parabolic equations which take the form $$u_t= \Delta u^m+ t^s| x|^\sigma u^p$$ with nonnegative initial value. Here $s\ge 0$, $m>(n-2)_+/n$, $p> \max(1,m)$ and $\sigma>-1$ if $n=1$ or $\sigma>-2$ if $n\ge 2$. We prove, among other things, that for $p\le pc$, where $p_c \equiv m+s (m-1)+ (2+2s+ \sigma)/n>1$, every nontrivial solution blows up in finite time. But for $p>p_c$ a positive global solution exists.

MSC 2000:: *35K65 Parabolic equations of degenerate type
35K15 Second order parabolic equations, initial value problems
35B40 Asymptotic behavior of solutions of PDE
35B05 General behavior of solutions of PDE

Keywords: nonnegative initial value; positive global solution

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