Global attractors for a chemotaxis system with superlinear growth in cross-diffusion rates and sub-logistic sources

This paper devotes to the study of dynamics for a chemotaxis system with sub-logistic sources in a two-dimensional domain. The existence result of global solutions of the considered system is established showing that incorporating sub-logistic sources can prevent blow-up solutions, even when the cross-diffusion rate exhibits superlinear growth. The proof is based on parabolic regularity in Sobolev and Orlicz spaces, combined with the variational method and the Moser iteration technique. Then, the asymptotic behavior of the solutions is further considered. For this the existence of a bounded absorbing set for the dynamical system is first proved through theory of evolution equations. It is ultimately obtained that the system possesses a global attractor.