## 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

### Keywords:

Global existence; uniform boundedness

Zbl 0702.35141
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