Global stability in “delay” systems with cross-diffusions dominated by self-diffusions

In this paper, applying a method of energy estimation similar to that of Seong-A. Shim [Seong-A. Shim, Uniform boundedness and convergence of solutions to the systems with cross-diffusions dominated by self-diffusions, Nonlinear Anal. RWA 4 (2003) 65–86], which used Gagliardo–Nirenberg-type inequalities in the estimates of solutions in order to establish W21-bounds uniform in time, we extend the result to the case of time delays, and establish conditions for the uniform boundedness of solutions and global asymptotic stability for the constant steady state of a quasilinear parabolic “delay” system with cross-diffusions dominated by self-diffusions and large diffusion coefficients in population dynamics.