On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

Abstract

We study nonlinear degenerate parabolic equations where the flux function f(x, t, u) does not depend Lipschitz continuously on the spatial location x. By properly adapting the "doubling of variables" device due to Kruikov and Carrillo, we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form k(x)f(u), where k(x) is a vector-valued function and f(u) is a scalar function.