Well-posedness for chemotaxis dynamics with nonlinear cell diffusion

This paper investigates the well-posedness of a reaction–diffusion system of chemotaxis type, with a nonlinear diffusion coefficient and a dynamics (growth–death) of the cell population b , and a stationary equation for the chemoattractant c . With respect to other works in which a nonlinear diffusion for cells has been considered, we treat here two distinct cases for this diffusion coefficient, the first in which it is a positive bounded function on R , and the other in which it may display a singularity at a finite value of the cell density. Essentially, the latter model is new and describes the saturation of the cell population in the neighborhood of a critical value for its diffusion coefficient. The chemotactic sensitivity function is supposed to depend both on the cell and chemoattractant densities. For homogeneous Neumann boundary conditions for the cell population and chemoattractant we prove the existence of a local in time solution when the L 2 -norm of the initial datum b 0 is sufficiently small and compute the maximum time interval for which the solution is bounded and smooth. Under a stronger assumption related to the chemotactic sensitivity we show that there exists a global in time solution for arbitrarily large initial data. In a case when the diffusion coefficient is singular, we focus on a model expressed by a variational inequality, describing the saturation of the cell population at the blowing-up diffusion value. Here, the proof requires the study of an intermediate problem with Robin boundary conditions, which may be interesting by itself. In all situations, uniqueness follows on a time interval included (not necessarily strictly) in that of the solution existence, under sufficient conditions.