Renormalized entropy solutions for degenerate nonlinear evolution problems

We study the degenerate differential equation $$ b(v)_t -hbox{ div}a(v, abla g(v))=f quad hbox{on }Q:= (0,T) imes Omega $$ with the initial condition $b(v(0,cdot))=b(v_0)$ on $Omega$ and boundary condition $v=u$ on some part of the boundary $Sigma:=(0,T) imes partial Omega$ 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 $A_1leq 0leq A_2$. Using Kruzhkov's method of doubling variables, we prove an existence and comparison result for renormalized entropy solutions.