Entropy solutions to a second order forward-backward parabolic differential equation

We prove that the first boundary value problem for a second order forward-backward parabolic differential equation in a bounded domain $G_T\subset\mathbb{R}^{d+1}$, where $d\geqslant2$, has a unique entropy solution in the sense of F. Otto. Under some natural restrictions on the boundary values this solution is constructed as the limit with respect to a small parameter of a sequence of solutions to Dirichlet problems for an elliptic differential equation. We also show that the entropy solution is stable in the metric of $L_1(G_T)$ with respect to perturbations of the boundary values in the metric of $L_1(\partial G_T)$.