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Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. (English) Zbl 0743.35038
The authors consider the problem \(u_ t=\Delta u-au^ p\), \(x\in\Omega\), \(t>0\), \({\partial u\over \partial n}=u^ q\), \(x\in\partial\Omega\), \(t>0\), \(u(x,0)=u_ 0(x)\geq 0\), \(x\in\overline\Omega\) with \(p,q>0\), \(a>0\), \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), \(u_ 0\not\equiv 0\). Global existence, uniform boundedness of solutions, properties of stationary solutions, convergence to stationary solutions and blow up are investigated, depending on conditions on the parameters \(p,q,a\) and the initial function \(u_ 0\). For \(n=1\) a complete answer is given, for \(n>1\) the answer is far from being complete. In an earlier work the second author considered the case \(a=0\) [e.g.: Commentat. Math. Univ. Carol. 30, No. 3, 479-484 (1989; Zbl 0702.35141)].
Reviewer: L.Simon (Budapest)

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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