Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals

Abstract

In this paper, we deal with the two-species chemotaxis-competition system with two signals under homogeneous Neumann boundary conditions in a general bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, by developing some Lp-estimate techniques, we show that the system possesses at least one global and bounded weak solutions, in addition to discussing the asymptotic behavior of the solutions. Our results generalize and improve previous results. As an application, we apply the existence results to consider the optimal control of the system we considered.