Non-autonomous lattice systems with switching effects and delayed recovery

The long term behavior of a type of non-autonomous lattice dynamical systems is investigated, where these have a diffusive nearest neighborhood interaction and discontinuous reaction terms with recoverable delays. This problem is of both biological and mathematical interests, due to its application in systems of excitable cells as well as general biological systems involving delayed recovery. The problem is formulated as an evolution inclusion with delays and the existence of weak and strong solutions is established. It is then shown that the solutions generate a set-valued non-autonomous dynamical system and that this non-autonomous dynamical system possesses a non-autonomous global pullback attractor.