On a regularity property and a priori estimates for solutions of nonlinear parabolic variational inequalities

Abstract

In this paper we consider the following nonlinear parabolic variational inequality; $u(t) \in D(\Phi)$ for all $t \in I$, $(u_t(t), u(t) - v) + \langle \Delta_p u(t), u(t) - v \rangle + \Phi(u(t)) - \Phi(v) \leqq (f(t), u(t) -v )$ for all $v \in D(\Phi)$ a.e. $t\in I$, $u(x,0) = u_0(x)$, where $\Delta_p$ is the so-called $p$-Laplace operator and $\Phi$ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data $u_0$ and $f$.