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Uniform bounds for solutions of a degenerate diffusion equation with nonlinear boundary conditions. (English) Zbl 0702.35142
The paper deals with solutions u(x,t) of the heat equation \(u_ t=\Delta u\) in the cylinder \(D\times (0,T)\), \(D\subset {\mathbb{R}}^ N\) bounded, under the nonlinear boundary condition \(\partial u/\partial v=f(u)\), \(f(u)u\leq L(| u|^{\alpha +1}+1)\), \(\alpha\geq 1\), \(L>0\). It is shown that there exist positive constants M,v (independent on T and \(u(\cdot,0))\) such that \[ \| u(\cdot,t)\|_{L^{\infty}(D)}\leq M(1+\| u(\cdot,0)\|_{L^{\infty}(D)})(1+\sup_{0\leq t\leq T}\oint_{\partial D}| u(s,t)|^ pds)^ v \] \(\forall t\in [0,T]\), whenever \(p>(N-1)(\alpha-1)\). It is proved that this estimate holds also for the equation \(((\beta(u))_ t=\Delta u\), where \(\beta (u)=| u|^{m-1}u\), \(0<m<\infty\), under the same nonlinear boundary condition. We do not expect that the exponent \((N-1)(\alpha-1)\) is sharp if \(m\neq 1\).
Reviewer: Ján Filo

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K05 Heat equation
35K55 Nonlinear parabolic equations
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