Global existence and uniqueness of solutions to a chemotaxis system

This paper is dealing with the existence and uniqueness of a global in-time bounded solution to a quasilinear chemotaxis system settled in a bounded domain under no-flux boundary conditions. The main difficulty occurs in the presence of the nonlinear logarithmic diffusion which can blow up around zero, and the energy estimates which do not provide a uniform estimate for large values of cell density u. Moreover, the chemo-sensitivity function depends on both cell and chemo-attractant densities which presents a further difficulty while proving the global existence of a solution. This is overcome by tracking the time evolution of a suitable functional. The existence of local in-time weak solutions is ensured using Schauder's fixed point theorem, while the uniqueness is obtained by adapting the method introduced in Diaz etal. [On a quasilinear degenerate system arising in semiconductor theory. Part I: existence and uniqueness of solutions. Nonlinear Anal. Real World Appl. 2001;2:305-336.]. Furthermore, under appropriate assumptions on the initial data, we prove that the solution is classical by using parabolic regularity results.