Existence of entropy measure-valued solutions for forward-backward $p$-parabolic equations

In this paper we have proved that the Dirichlet problem for the forward-backward $p$-parabolic equation has an entropy measure-valued solution which has been obtained as a singular limit of weak solutions and their gradients to the Dirichlet problem for the elliptic equation containing the anisotropic $(p,2)$-Laplace operator. In order to guarantee the existence of entropy measure-valued solutions, the initial and final conditions should be formulated in the form of integral inequalities. This means that an entropy measure-valued solution can deviate from both initial and final data on the boundary. Moreover, a gradient Young measure appears in the representation of an entropy measure-valued solution. The uniqueness of entropy measure-valued solutions is still an open question.