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Existence, nonexistence and some asymptotic behaviour of global solutions of a nonlinear degenerate parabolic equation. (English) Zbl 0563.35038
Let be $$T>0$$ and let $$\Omega \subset R^ n$$ be a bounded domain with smooth boundary $$\partial \Omega$$. The author considers the existence, nonexistence and some asymptotic behavior of global weak solutions of the problem $(1)\quad u_ t-\Delta u^{p+1}-u^{\alpha -1}=0\quad in\quad \Omega \times [0,T]$
$(2)\quad u(x,0)=u_ 0\quad for\quad x\in \Omega$
$(3)\quad u(x,t)|_{\partial \Omega}=0\quad for\quad t\in [0,T]$
$(4)\quad u\geq 0,$ where $$\alpha >0$$, $$p>0$$. Roughly speaking the results are stated as follows: if $$p>\alpha$$ the problem (1)-(4) has a global solution for each $$u_ 0(x)\geq 0$$, $$| u_ 0|^ pu_ 0\in H^ 0_ 1(\Omega)$$ and $$p<\alpha$$ the global existence and nonexistence depends on the initial data. Related results are obtained in a paper by V. A. Galaktionov [Differ. Uravn. 17, 836-842 (1981; Zbl 0468.35056)].
Reviewer: J.Goncerzewicz

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K65 Degenerate parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations