The global asymptotic behavior in some functional parabolic equations was presented. The sufficient condition for the global convergence to a spatially uniform equilibrium in the system of functional partial differential equations was provided. Results were observed to be valid for all non-negative spatio-temporal kernels, which also satisfied a uniform normailzation condition.
Laister, R. (2002). Global asymptotic behaviour in some functional parabolic equations. Nonlinear Analysis: Theory, Methods and Applications, 50(3), 347-361. https://doi.org/10.1016/S0362-546X%2801%2900766-0