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The critical exponents of parabolic equations and blow-up in ${R}^{n}$. (English) Zbl 0892.35088

Summary: We study the Cauchy problem in ${ℝ}^{n}$ of general parabolic equations which take the form

${u}_{t}={\Delta }{u}^{m}+{t}^{s}{|x|}^{\sigma }{u}^{p}$

with nonnegative initial value. Here $s\ge 0$, $m>{\left(n-2\right)}_{+}/n$, $p>max\left(1,m\right)$ and $\sigma >-1$ if $n=1$ or $\sigma >-2$ if $n\ge 2$. We prove, among other things, that for $p\le pc$, where ${p}_{c}\equiv m+s\left(m-1\right)+\left(2+2s+\sigma \right)/n>1$, every nontrivial solution blows up in finite time. But for $p>{p}_{c}$ a positive global solution exists.

##### MSC:
 35K65 Parabolic equations of degenerate type 35K15 Second order parabolic equations, initial value problems 35B40 Asymptotic behavior of solutions of PDE 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
##### Keywords:
nonnegative initial value; positive global solution