Solvability of doubly nonlinear parabolic equation with p -laplacian

In this paper, we consider a doubly nonlinear parabolic equation \begin{document}$ \partial _t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f $\end{document} with the homogeneous Dirichlet boundary condition in a bounded domain, where \begin{document}$ \beta : \mathbb{R} \to 2 ^{ \mathbb{R} } $\end{document} is a maximal monotone graph satisfying \begin{document}$ 0 \in \beta (0) $\end{document} and \begin{document}$ \nabla \cdot \alpha (x , \nabla u ) $\end{document} stands for a generalized \begin{document}$ p $\end{document} -Laplacian. Existence of solution to the initial boundary value problem of this equation has been studied in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on \begin{document}$ \beta $\end{document} . However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for \begin{document}$ 1 . Main purpose of this paper is to show the solvability of the initial boundary value problem for any \begin{document}$ p \in (1, \infty ) $\end{document} without any conditions for \begin{document}$ \beta $\end{document} except \begin{document}$ 0 \in \beta (0) $\end{document} . We also discuss the uniqueness of solution by using properties of entropy solution.