Global dynamics for a class of reaction-diffusion equations with distributed delay and Neumann condition

In this paper, we investigate a class of non-monotone reaction-diffusion equations with distributed delay and a homogenous boundary Neumann condition, which have a positive steady state. The main concern is the global attractivity of the unique positive steady state. To achieve this, we use an argument of a sub and super-solution combined with fluctuation method. We also give a condition for which the exponential stability of the positive steady state is reached. As an example, we apply our results to diffusive Nicholson blowflies and diffusive Mackey-Glass equation with distributed delay. We point out that we obtain some new results on exponential stability of the positive steady state for these cited models.