On some quasilinear parabolic equations with non-monotone multivalued terms

Abstract

In this paper, we study the existence of solutions to the initial-boundary value problem for the following parabolic differential inclusion: \begin{document}$ \begin{equation*} \begin{cases} u_t\left(t, x \right) -\triangle _{p}u\left( t, x \right) \in -\partial \phi \left( u\left( t, x \right) \right) + G\left( t, x, u\left( t, x \right) \right) & (t, x) \in Q_T, \\ u(t, x) = 0 & (t, x) \in \Gamma_T, \\ u(0, x) = u_0(x) & x \in \Omega, \end{cases} \end{equation*} $\end{document} where $ \Omega $ is a bounded open subset of $ \mathbb{R}^{N} $ with smooth boundary $ \partial \Omega, $ $ T>0 $, $ Q_{T}: = [0, T] \times \Omega $, $ \Gamma_T: = [0, T] \times \partial\Omega $, $ u_t = \frac{\partial u}{\partial t} $, $ \triangle_{p} $ is the $ p $-Laplace differential operator, $ \partial \phi $ denotes the subdifferential of a proper lower semicontinuous convex function $ \phi :\mathbb{R}\rightarrow \left[ 0, \infty \right] $, and $ G:Q_{T}\times \mathbb{R}\rightarrow 2^{ \mathbb{R}} \backslash \{\emptyset\} $ is a nonmonotone multivalued mapping.The case where $ \phi \equiv 0 $ and $ G(t, x, u) = |u|^{q-2}u $ gives the prototype of our problem, denoted by (E)$ _p $. The existence of time-local strong solutions for (E)$ _p $ is already studied by several authors. However, these results require a stronger assumption on $ q $ than that for the semi-linear case (E)$ _p $ with $ p = 2 $.More precisely, it has been long conjectured that (E)$ _p $ should admit a time-local strong solution for the Sobolev-subcritical range of $ q $, i.e., for all $ q \in (2, p^\ast) $ with $ p^\ast = \infty $ for $ p \geq N $ and $ p^\ast = \frac{N p}{N-p} $ for $ p<N $, which is the well-known fact for the semi-linear case (E)$ _p $ with $ p = 2 $.The main purpose of the present paper is to show this conjecture holds true and to extend this classical study to the cases where $ u \mapsto G(\cdot, \cdot, u) $ is upper semicontinuous or lower semicontinuous, each one is a generalized notion of the continuity in the theory of multivalued analysis.We also discuss the extension of large or small local solutions along the lines of arguments developed in [28].