Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities

The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term ∂t u is replace by ∂t b (u), with memory terms and quasilinear variational inequalities is shown. The existence of solutions of equations is proved under the assumption that the nonlinear function b is monotone and a gradient of a convex, continuously differentiable function. The uniqueness is proved for Lipschitz-continuous elliptic parts. The existence of solutions of quasilinear variational inequalities is proved under stronger assumptions, namely, the nonlinear function defining the elliptic part is assumed to be a gradient and the function b to be Lipschitz continuous.