Vasile Staicu
University of Aveiro, Aveiro, Portugal

The aim of this talk is to present some recent joint results with Mitsuharu Otani concerning the local and the global existence of strong solutions to the following parabolic differential inclusion in $Q_{T}:=[0,T]\times \Omega $: $$ \displaystyle\frac{\partial }{\partial t}u\left( t,x\right) -\triangle _{p}u\left( t,x\right) \in -\partial \phi \left( u\left( t,x\right) \right) +G\left( t,x,u\left( t,x\right) \right) , $$ where $\Omega $ is a bounded open subset of $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega ,$ $T>0$, $\triangle _{p}$ 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.

We firstly set up a framework which enables us to treat wider nonlinearity of $G(\cdot ,\cdot ,u)$, more precisely, to cover the growth condition on $G(\cdot ,\cdot ,u)$ up to the Sobolev-subcritical range, and secondly we adapt and improve the techniques and arguments developed in [3] and [4], in order to obtain existence results for the initial boundary value problem to parabolic inclusion generalizing corresponding results given by many authors, especially given in [1], [2] and [4] where the semi-linear case $ p=2$ is considered.

We give two types of local existence results for the cases where $G(\cdot ,\cdot ,u)$ is upper semicontinuous and lower semicontinuous, and also discuss the extension of large or small local solutions along the lines of arguments developed in [1].

[1] M. Otani, Non-monotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46:268-299, 1982
[2] M. Otani, Non-monotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Periodic problems, J. Differential Equations, 54:248-273, 1984
[3] M. Otani, V. Staicu, Existence results for quasilinear elliptic equations with multivalued nonlinear terms, Set-Valued Var. Anal., 22:859-877, 2014
[4] M. Otani, V. Staicu, On some nonlinear parabolic equations with nonmonotone multivalued terms, J. Convex Anal., vol. 28, No. 3, 2021