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A critical exponent in a degenerate parabolic equation. (English) Zbl 1007.35043
The global existence of positive solutions of the equation $$u_t=u^p \Delta u+u^q$$ in $${\mathbb R}^n$$ is studied for $$p,q\geq 1$$. It is shown that all positive solutions are global but unbounded if $$1\leq q<p+1$$ ($$1\leq q<3/2$$ if $$p=1$$) and the initial function $$u_0$$ decays to zero rapidly enough as $$|x|\to\infty$$. On the other hand, for $$q=p+1$$ all positive solutions blow up in finite time. Both global and nonglobal solutions exist if $$q>p+1$$.

##### MSC:
 35K65 Degenerate parabolic equations 35B33 Critical exponents in context of PDEs
##### Keywords:
global existence; blow-up
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##### References:
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