Solutions of the attraction-repulsion-chemotaxis system with nonlinear diffusion

In this study, we consider the well-posedness of the attraction-repulsion chemotaxis system. This paper explores the dynamics of species movement in reaction to two chemically opposing substances, incorporating nonlinear diffusion. Our primary objective is to establish the existence of a global-in-time weak solution for the proposed model in an unbounded three-dimensional spatial domain. Our study has confirmed the existence of a global-in-time weak solution for the proposed system in three dimensions. Furthermore, we demonstrate that global-in-time weak solutions are also attainable for the proposed system in a bounded domain with a smooth boundary.