Existence and uniqueness of a weak solution to the initial mixt boundary value problem for quasilinear elliptic-parabolic equations

We prove the existence and the uniqueness of a weak solution to the mixed boundary problem for the elliptic-parabolic equation
$$\begin{gather*} \partial_tb(u)-\operatorname{div}\{|\sigma(u)|^{m-2}\sigma(u)\}=f(x,t), \\ \delta(u):=\nabla u+k(b(u))\vec e, \qquad |\vec e|=1, \enskip m>1, \end{gather*}$$
with a monotone nondecreasing continuous function $b$. Such equations arise in the theory of non-Newtonian filtration as well as in the mathematical glaciology.