Existence and uniqueness of regular solution of Cauchy–Dirichlet problem for some class of doubly nonlinear parabolic equations

The class of equations of the type
\begin{equation} \partial u/\partial t-\operatorname{div}\vec a(u,\nabla u)=f, \tag{1} \end{equation}
such that
\begin{equation} \begin{gathered} \vec a(u,p)\cdot p\ge\nu_0|u|^l|p|^m-\Phi_0(u),\\ |\vec a(u,p)|\le\mu_1|u|^l|p|^{m-1}+\Phi_1(u) \end{gathered} \tag{2} \end{equation}
with some $m\in(1,2)$, $l\ge0$ and $\Phi_i(u)\ge0$ is studied. Similar equations arise in the study of turbulent filtration of gas or a liquid through porous media. Existence and uniqueness in some class of Hölder continuous generalized solutions of Cauchy–Dirichlet problem for equations of the type (1), (2) is proved. Bibliography: 9 titles.