Dissipative Gradient Nonlinearities Prevent δ $\delta$ -Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models

Abstract

We study a class of zero-flux attraction–repulsion chemotaxis models, characterized by nonlinearities laws for the diffusion of the cell density $u$, the chemosensitivities and the production rates of the chemoattractant $v$ and the chemorepellent $w$. In addition, a source involving also the gradient of $u$ is incorporated. Our overall study touches on different aspects: we address questions connected to local well-posedness, we derive sufficient conditions to ensure boundedness of solutions, and finally, we develop numerical simulations giving insights into the evolution of the system.