Asymptotic behavior of solutions to parabolic-elliptic variational inequalities

Abstract

in a real Hilbert space H, where J is a given interval in R, = [0, co), &#J’ is the subdifferential of a convex function 4’ on H, and B is a maximal monotone operator in H. The abstract setting reflects the form of a class of parabolic-elliptic boundary value problems with time-dependent obstacles. Such problems arise, in particular, as mathematical models of various processes in electrochemical technology and some flows with saturations in porous media. Time-dependent obstacles may represent geometric controls (moving control actions) in these problems. In particular, such controls may be comprehended in variable boundary conditions prescribed over time-dependent parts of the boundary of domain or in source terms distributed over time-dependent subdomains. The former refers to control of flows via variable water table in reservoirs, the latter characterizes control of electrochemical process via mechanically driven motion of a toolpiece (cathode). The most typical control objective consists there in stabilizing the state of the process at a given spatial distribution (for example constant water head and desired shape of anode). This motivates our interest in the asymptotic behavior of solutions to the corresponding boundary value problems as t +a. To the knowledge of the authors, this question has so far not been studied in literature. As it has been shown in Kenmochi-Pawlow [16, 171, the class of considered time-dependent problems admits a representation in the form of evolution equation E(#, B). In [16, 171, the corresponding results on existence, uniqueness and stability of solutions have been established. In the present paper we study the asymptotic properties of solutions to E(c$‘, B) as t + 00, provided some hypotheses are imposed on the family ($‘; t E R,) and the limit 4” of # as t -+ a. For the solution (u, u*) of E(d’, B) on R, we prove that