Global boundedness of solutions to a class of partial differential equations with time delay

A class of diffusive partial differential equations with strongly coupled time delays and diffusion is considered. The global boundedness of weak solutions of the equation is proved by an entropy method that was initially proposed for studying the global boundedness of reaction-diffusion equations with cross-diffusion. The presence of the time delays in the equation prevents the entropy method to be directly applied, and here we extend the entropy method to this class of diffusive partial differential equations with time delays by proving some key entropy inequalities, which further allows us to obtain the estimates of gradient of the solutions. The results can be used to show the global boundedness of solutions of population models with memory effect, which were recently proposed for describing the movement of highly-developed animal species. In addition, we show that the results are also applicable for the classic partial functional differential equations, where the time delays only appear in the reaction terms.