Solvability of doubly nonlinear parabolic equation with <i>p</i>-laplacian

Abstract

&lt;p style='text-indent:20px;'&gt;In this paper, we consider a doubly nonlinear parabolic equation &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \partial _t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with the homogeneous Dirichlet boundary condition in a bounded domain, where &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \beta : \mathbb{R} \to 2 ^{ \mathbb{R} } $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a maximal monotone graph satisfying &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ 0 \in \beta (0) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \nabla \cdot \alpha (x , \nabla u ) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; stands for a generalized &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Laplacian. Existence of solution to the initial boundary value problem of this equation has been studied in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \beta $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ 1 &amp;lt; p &amp;lt; 2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Main purpose of this paper is to show the solvability of the initial boundary value problem for any &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ p \in (1, \infty ) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; without any conditions for &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ \beta $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; except &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ 0 \in \beta (0) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. We also discuss the uniqueness of solution by using properties of entropy solution.&lt;/p&gt;