Existence and uniqueness for a degenerate parabolic equation with 𝐿¹-data

Andreu, F.; Mazón, J.; de León, S.; Toledo, J.

doi:10.1090/S0002-9947-99-01981-9

Existence and uniqueness for a degenerate parabolic equation with $L^1$-data
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by F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo PDF

Trans. Amer. Math. Soc. 351 (1999), 285-306 Request permission

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} \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 $\mathbf {a}$, $Du$ 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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