Electron. J. Diff. Equ., Vol. 2016 (2016), No. 77, pp. 1-11.

Entropy solutions of exterior problems for nonlinear degenerate parabolic equations with nonhomogeneous boundary condition

Li Zhang, Ning Su

In this article, we consider the exterior problem for the nonlinear degenerate parabolic equation
 u_t - \Delta b(u) + \nabla \cdot \Phi(u) = F(u),
 \quad (t,x) \in (0,T) \times \Omega,
$\Omega$ is the exterior domain of $\Omega_0$ (a closed bounded domain in $\mathbb{R}^N$ with its boundary $ \Gamma \in \mathcal{C}^{1,1}$), $b$ is non-decreasing and Lipschitz continuous, $\Phi=(\phi_1,\dots,\phi_N)$ is vectorial continuous, and F is Lipschitz continuous. In the nonhomogeneous boundary condition where $b(u) = b(a)$ on $(0,T) \times \Gamma$, we establish the comparison and uniqueness, the existence using penalized method.

Submitted January 29, 2015. Published March 18, 2016.
Math Subject Classifications: 35K55, 35K65.
Key Words: Degenerate parabolic equation; exterior problem; nonlinear; entropy solution.

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Li Zhang
Management School
Hangzhou Dianzi University
Hangzhou 310018, China
email: zhli25@163.com
Ning Su
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China
email: nsu@math.tsinghua.edu.cn

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