Large time behavior in a reaction diffusion epidemic model with logistic source

Abstract

This paper concerns a reaction diffusion epidemic model under logistic source. First, we explore uniform boundedness of the system. Moreover, by constructing two Lyapunov functions, we show that the disease-free equilibrium of system is globally asymptotically stable and the large time dynamics of constant endemic equilibrium is also considered. Especially, we further investigate the convergence rate of the nonnegative globally bounded classical solution. Finally, two numerical simulations are displayed to confirm the results.