Global dynamics for a class of non-monotone time-delayed reaction-diffusion equations

This paper deals with a large class of non-monotone time-delayed reaction-diffusion equations in which the reaction term can be spatially nonlocal. Nonexistence, existence, uniqueness and global attractivity of positive equilibriums to the equation are addressed. In particular, developed is a technique that combines the method of super-sub solutions, the variation-of-constants formula for the delay differential equation and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the global attractiveness of the unique positive equilibrium. Two examples are given to illustrate the obtained results.