The well-posedness of renormalized solutions for a non-uniformly parabolic equation

In this paper we present a unified approach to establish the existence of renormalized solutions and a comparison result for a class of nonuniformly parabolic initial-boundary value problems. As a consequence, the uniqueness of renormalized solutions and the equivalence between entropy and renormalized solutions for such equations are obtained. The results extend the well-posedness results for the classical p-Laplacian type equations to a larger class of non-linear elliptic and parabolic PDEs including the nearly linear growth operators.