On the solvability of a nonlinear evolutionary inequality of the theory of joint motion of surface and ground waters (Q1390380)

We study an evolutionary variational inequality degenerating on both the solution and its gradient. Such inequalities arise in the modeling of the joint motion of surface and ground waters when the process is considered in a domain with a slit on which an additional condition is given in the form of one-dimensional partial differential either equation or inequality. Under sufficiently general assumptions on the smoothness of the initial data, the existence of a non-negative solution in a class of generalized functions is proved by a semidiscretization method with penalty. We show that a sequence of solutions of the penalized semidiscrete problems converges to the solution of the original inequality.