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Existence and uniqueness for a degenerate parabolic equation with $L^{1}$-data

Author(s): F. Andreu; J. M. Mazón; S. Segura de León; J. Toledo
Journal: Trans. Amer. Math. Soc. 351 (1999), 285-306.
MSC (1991): Primary 35K65, 47H20
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Abstract: In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in $L^{1}(\Omega )$,

\begin{equation*}u_{t} = \mathrm{div} {\hbox{$\mathbf a$}}(x,Du) \quad \text{in } (0, \infty ) \times \Omega, \end{equation*}

\begin{equation*}-{\frac{{\partial u} }{{\partial \eta _{a}}}} \in \beta (u) \quad \text{on } (0, \infty ) \times \partial \Omega,\end{equation*}

\begin{equation*}u(x, 0) = u_{0}(x) \quad \text{in }\Omega ,\end{equation*}

where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, $\partial / {\partial \eta _{a}}$ is the Neumann boundary operator associated to ${\hbox{$\mathbf a$}}$, $Du$ the gradient of $u$ and $\beta $ is a maximal monotone graph in ${\mathbb{R}}\times {\mathbb{R}}$ with $0 \in \beta (0)$.

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Additional Information:

F. Andreu
Affiliation: Departamento de Análisis Matemático, Universitat de València, 46100 Burjassot, Valencia, Spain
Email: Fuensanta.Andreu@uv.es

J. M. Mazón
Affiliation: Departamento de Análisis Matemático, Universitat de València, 46100 Burjassot, Valencia, Spain
Email: Mazon@uv.es

S. Segura de León
Affiliation: Departamento de Análisis Matemático, Universitat de València, 46100 Burjassot, Valencia, Spain
Email: Sergio.Segura@uv.es

J. Toledo
Affiliation: Departamento de Análisis Matemático, Universitat de València, 46100 Burjassot, Valencia, Spain
Email: Jose.Toledo@uv.es

DOI: 10.1090/S0002-9947-99-01981-9
PII: S 0002-9947(99)01981-9
Keywords: Non-linear parabolic equations, non-linear boundary conditions, $p$-Laplacian, accretive operators, mild solutions
Received by editor(s): September 11, 1995
Received by editor(s) in revised form: December 2, 1996
Additional Notes: This research has been partially supported by DGICYT, Project PB94-0960
Copyright of article: Copyright 1999, American Mathematical Society

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