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

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.

Keywords

• Degenerate parabolic equation
• exterior problem
• nonlinear
• entropy solution