On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients (Q1811365)

The authors study initial value problems of the form \[ u_t + \text{div}f(x,t,u) = \Delta A(u) + q(x,t,u),\quad (x,t)\in \Pi_T = \mathbb{R}^d\times (0,T), \] where \(T > 0\) is fixed, \(u(x,t)\) is the scalar unknown function. The coefficients \(f\), \(A\), \(q\) of the above-stated problem satisfy appropriate regularity assumptions and \[ A\in \text{Lip}_{\text{loc}}(\mathbb{R}),\quad A(\cdot) \quad \text{is nondecreasing with } A(0) = 0. \] The main aim is to prove a uniqueness result within the class of entropy solutions for the initial value problem and to expose a result concerning the continuous dependence on the initial data and flux function of the form \(k(x)f(u)\), where \(k(x)\) is a vector-valued function and \(f(u)\) is a scalar function.