Periodic-parabolic logistic equation with mixed local and nonlocal diffusion

Abstract This work investigates a class of periodic-parabolic nonlocal operators and their application to nonlinear evolution problems. We present a characterization of the Maximum Principle in terms of the spectral properties of the associated operator, and we analyze the approximation of its principal eigenvalue using the theory of evolution systems. As an application, we establish the existence and uniqueness of a positive solution for the logistic equation driven by this nonlocal operator. In addition, we derive several results regarding the blow-up phenomenon.