On a regularity property and a priori estimates for solutions of nonlinear parabolic variational inequalities (Q2709968)

The author considers the following nonlinear parabolic variational inequality with mixed boundary conditions: NEWLINE\[NEWLINE\begin{aligned} & u(t)\in D(\Phi) \text{ for all }t\in I:\\ & \bigl(u_t(t),u(t)-v\bigr)+ \langle\Delta_p u(t)-v \rangle+ \Phi\bigl(u(t) \bigr)- \Phi(v) \leq\bigl(f(t), u(t)-v\bigr)\text{ for all }v\in D(\Phi),\text{ a.e. }t\in I,\\ & u(x,0)= u_0(x),\end{aligned}NEWLINE\]NEWLINE where \(\Delta_p\) is the \(p\)-Laplace operator and \(\Phi\) is a proper, lower semicontinuous functional. Regularity properties of solutions and the continuous dependence of solutions on given data \(u_0\) and \(f\) are proved.