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# On some quasilinear parabolic equations with non-monotone multivalued terms

• *Corresponding author: Vasile Staicu

Dedicated to Professor Viorel Barbu on the occasion of his 80th birthday

The first author is partially supported by the Grant-in-Aid for Scientific Research, #18K03382, the Ministry of Education, Culture, Sports, Science and Technology, Japan.
The second author is partially supported by Portuguese funds through CIDMA and the Portuguese Foundation for Science and Technology (FCT), within the project UID/MAT/04106/2013

• In this paper, we study the existence of solutions to the initial-boundary value problem for the following parabolic differential inclusion:

$\begin{equation*} \begin{cases} u_t\left(t, x \right) -\triangle _{p}u\left( t, x \right) \in -\partial \phi \left( u\left( t, x \right) \right) + G\left( t, x, u\left( t, x \right) \right) & (t, x) \in Q_T, \\ u(t, x) = 0 & (t, x) \in \Gamma_T, \\ u(0, x) = u_0(x) & x \in \Omega, \end{cases} \end{equation*}$

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 .

Mathematics Subject Classification: Primary: 35K20, 35A16, 35K92, 35R70, 35D35.

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