Well-posedness, monotonicity, asymptotic translation of the Cauchy problem of delayed reaction–diffusion systems

In this paper, we establish a new basic theory on the Cauchy problem of delayed reaction–diffusion systems on the whole Euclidean space, where the initial function space is equipped with the compact open topology (also called coarse topology). Generally, under this coarse topology, the reaction terms are not locally Lipschitz continuous. Even they are, it is difficult to boil down the basic theory on the Cauchy problem to the (parameterized) contraction mapping problem. To overcome this difficulty, we explain and prove the uniqueness of solutions from a new perspective. This provides a unified way to obtain the basic theory on the Cauchy problem of delayed reaction–diffusion systems on the whole Euclidean space, which includes the local existence, uniqueness, and continuous dependence of solutions. Moreover, we show that, with respect to the compact open topology, the solution semiflow is monotone and possesses the property of asymptotic translation.