Invariance and strict invariance for nonlinear evolution problems with applications

Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) m-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraining functional are expressed in terms of the Dini derivative of that functional, outside the considered set in directions determined by the considered m-accretive operator. An approach for non-reflexive Banach spaces is developed and some result improving a recent paper (Cannarsa et al., 2018) is presented. Applications to nonlinear obstacle problems and age-structured population models are presented in spaces of continuous functions where an advantage of such approach is taken. Moreover, some new abstract criteria for the so-called strict invariance are derived and their direct application to problems with barriers is discussed.