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

Abstract:
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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