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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

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