Parabolic--Elliptic Dynamics with Local--Nonlocal Coupled Operators

Abstract

In this paper, we study two local–nonlocal settings for parabolic–elliptic evolution systems. In our problems we have a disjoint partition of the spacial domain Ω as Ω = A ∪ B and we first consider a local parabolic equation posed in A with a nonlocal elliptic balance equation acting in the complementary subdomain B. Next, we reverse the roles and take a local elliptic equation posed in A coupled with a nonlocal parabolic equation acting in B. In both models, the interaction between the two regions is driven by a nonlocal transmission term given by a kernel that transfers mass across the interface, giving rise to a mixed local–nonlocal, elliptic–parabolic dynamics. We consider Neumann boundary conditions for both problems. To being our analysis we first establish the existence and uniqueness of solutions using a fixed point argument. Then, we provide a detailed analysis of their qualitative behavior. In particular, we show that the coupling structure induces a natural energy functional whose gradient flow governs the evolution, despite the elliptic–parabolic nature of the system. As it is expected in Neumann settings, we prove that the total mass in the whole domain Ω is preserved in time. We also analyze the long-time behaviour and obtain decay estimates for the parabolic component, which in turn drive the convergence of the elliptic part to a constant solution. Finally, we prove that the parabolic–elliptic problem under consideration is the limit of a purely parabolic problem when a parameter that controls the speed of the dynamic at which one component evolves goes to zero.