Reaction diffusion equation with spatio-temporal delay

• One of our main work is to study the initial value problem of reaction–diffusion equation with spatio-temporal delay.
• By the maximal, minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability.
• Applying these theories to Lotka–Volterra model with spatio-temporal delay.

We investigate reaction–diffusion equation with spatio-temporal delays, the global existence, uniqueness and asymptotic behavior of solutions for which in relation to constant steady-state solution, included in the region of attraction of a stable steady solution. It is shown that if the delay reaction function satisfies some conditions and the system possesses a pair of upper and lower solutions then there exists a unique global solution. In terms of the maximal and minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability of reaction–diffusion equation with spatio-temporal delay. Applying this theory to Lotka–Volterra model with spatio-temporal delay, we get the global solution asymptotically tend to the steady-state problem’s steady-state solution.