Nonlocal quasivariational evolution problems
This is a valuable paper indeed. The existence of the evolution problem \[ u'(t)+\partial\varphi(u,t,u(t))\ni 0 \] with \(u(0)=u_0\) in a Hilbert space \(H\) is studied. We emphasize the functional dependence of \(\varphi\) on \(u\) in order to take into account nonlocal in time effects. The function \(\varphi\) is convex in its last variable and \(\partial\) stands for the subgradient with respect to it. The existence of weak solutions relies on an order structure and on a fixed point theorem for multivalued maps on ordered sets. Various examples are included: Moreau's sweeping process, quasivariational sweeping process, gradient flow, parabolic variational inequalities, parabolic evolution problems.
scientific article; zbMATH DE number 5078199
scientific article