Global well-posedness and dynamics of spatial diffusion HIV model with CTLs response and chemotaxis

In this paper, we study the global well-posedness and global dynamics of a reaction–diffusion HIV infection model with the chemotactic movement of CTLs (Cytotoxic T lymphocytes). We first show the global existence and uniform boundedness for solutions of the system with general functional incidences. Then, for the model with bilinear incidence rate, we discuss the existence conditions of the three equilibria (infection-free, chemokines-extinct, chemokines-acute equilibria) of the model and obtain the conclusion of the local asymptotic stability of these equilibria by analyzing the linearized system at these equilibria. Moreover, by constructing reasonable Lyapunov functionals, we investigate the global stability and attractivity of the equilibria. Applying the $L^p-L^q$ estimate, Young’s inequality, Gagiardo-Nirenberg inequality and the parabolic regularity theorem, we also discuss the convergence rates of the equilibria. Finally, some numerical simulations are conducted to verify the theoretical results.