Trapping regions for discontinuously coupled systems of evolution variational inequalities and application

We consider initial-boundary value problems for weakly coupled systems of parabolic equations under coupled nonlinear flux boundary condition. Both coupling vector fields f: Q × ℝ² →² and g: Γ × ℝ² → ℝ² are assumed to be either of competitive or cooperative type, but may otherwise be discontinuous with respect to all their arguments. The main goal is to provide conditions for the vector f and g that allow the identification of regions of existence of solutions (so called trapping regions). To this end the problem is transformed to a discontinuously coupled system of evolution variational inequalities. Assuming a generalized outward pointing vector field on the boundary of a rectangle of the dependent variable space, the system of evolution variational inequalities is solved via a fixed point problem for some increasing operator in an appropriate ordered Banach space. The main tools used in the proof are evolution variational inequalities, comparison techniques, and fixed point results in ordered Banach spaces.