Doubly coupled systems of parabolic hemivariational inequalities: Existence and extremal solutions

Abstract

We consider an initial–boundary value problem for a quasilinear parabolic system of hemivariational inequalities which is not necessarily coercive. The system exhibits full dependence on the gradient of the solution and is doubly coupled on both the source and multivalued terms. Based on sub-supersolutions, truncation functions, and nonsmooth analysis we establish an existence and enclosure result for weak solutions within a trapping region. The existence of extremal solutions is proved without imposing any monotonicity conditions on lower order terms. Moreover, we present a sufficient condition to construct effectively a trapping region and show the existence of positive solutions.