Global asymptotic behavior of solutions for a parabolic-parabolic-ODE chemotaxis system modeling multiple sclerosis

Abstract

In this paper, we focus on a chemotaxis model of multiple sclerosis, which is proposed by Lombardo et al. (2017) [10]. The model consists of a chemotaxis quasi-linear parabolic equation describing the evolution of the density of activated macrophages, a semilinear parabolic equation ruling the evolution of the concentration of cytokine and an ordinary different equation modeling the evolution of the density of apoptotic oligodendrocytes. In this model, activated macrophages stimulated by a cytokine move along the concentration of cytokine gradient, which is termed chemotaxis. Our first purpose is to rigorously prove the boundedness and global existence of the classical solution to this model by applying heat operator semigroup theory and establishing some a priori estimates. The another purpose is to study the global asymptotic stability of a unique positive equilibrium, which together with the Turing instability are two aspects characteristic of mutual complement.