Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 147, pp. 1-32.

Title: Renormalized entropy solutions for degenerate nonlinear 
       evolution problems  

Author: Kaouther Ammar (Technische Univ. Berlin, Germany)

Abstract:
  We study the degenerate differential equation
 $$
  b(v)_t -\hbox{ div}a(v,\nabla g(v))=f \quad
 \hbox{on }Q:= (0,T) \times \Omega
 $$
 with the initial condition $b(v(0,\cdot))=b(v_0)$ on $\Omega$ and
 boundary condition $v=u$ on some part of the boundary
 $\Sigma:=(0,T) \times \partial \Omega$ with  $g(u)\equiv 0$ a.e. on
 $\Sigma$. The vector field $a$ is
 assumed to satisfy the Leray-Lions conditions, and the
 functions  $b,g$ to be continuous, locally Lipschitz, nondecreasing
 and to satisfy the normalization condition $b(0)=g(0)=0$ and the
 range condition $R(b+g)=\mathbb{R}$.
 We assume also that $g$ has a  flat region $[A_1,A_2]$
 with $A_1\leq 0\leq A_2$. Using Kruzhkov's method of doubling
 variables, we prove an existence and  comparison result
 for renormalized entropy solutions.

Submitted August 15, 2009. Published November 20, 2009.
Math Subject Classifications: 35K55, 35J65, 35J70, 35B30.
Key Words: Renormalized; degenerate; diffusion;
          homogenous boundary conditions; continuous flux.