Existence of solutions of parabolic variational inequalities with one-sided restrictions

We prove sufficient conditions for the existence of a solution of a “strong” nonlinear variational inequality of parabolic type. The theory can be used for solving parabolic equations with one-sided boundary conditions. As an example, we prove the existence of a solution of a strong parabolic variational inequality with $p$-Laplacian in the Sobolev space $L_p(0,T;W_p^1(\Omega))$, $p\in[2,\infty)$.