Dissipative Gradient Nonlinearities Prevent δ-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 (Formula presented.), the chemosensitivities and the production rates of the chemoattractant (Formula presented.) and the chemorepellent (Formula presented.). In addition, a source involving also the gradient of (Formula presented.) 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.