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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

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