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Zbl 0778.35046
Blanchard, D.; Francfort, G.A.
A few results on a class of degenerate parabolic equations.
(English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18, No.2, 213-249 (1991). ISSN 0391-173X

The authors study the solvability of the possibly degenerate parabolic problem $${\partial\over\partial t}b(u)-\text{div} D\varphi(\text{grad} u)=f\quad\text{in }\Omega\times(0,T),$$ $$u=0\text{ on }\partial\Omega\times(0,T),\quad b(u)\vert\sb{t=0}=b(u\sb 0)\text{ in }\Omega,$$ where $\Omega\subset\bbfR\sp N$ is a bounded domain, $b$ is monotone and real valued, and $\varphi$ is a convex, coercive potential. Despite its rather innocuous title, this paper contains a number of important, new and deep results, and it extends and complements previous work of the authors as well as work of Alt and Luckhaus and others. The theorems are too technical to repeat in detail here. Let us simply note that particular care is paid to a careful formulation of the hypotheses on $b,f,\varphi$ and $\Omega$. Existence results are carefully stated and the theorems are proven in a Sobolev space setting. Comparison results are also proven.
[R.Guenther (Corvallis)]
MSC 2000:
*35K20 Second order parabolic equations, boundary value problems
35K60 (Nonlinear) BVP for (non)linear parabolic equations

Keywords: parabolic boundary value problems; existence; comparison results; Sobolev space

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