Global existence and uniqueness of solutions to a chemotaxis system

ABSTRACT

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 et al. [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.

COMMUNICATED BY:

• Roger Temam

Keywords:

• Nonlinear diffusion
• chemotaxis
• Keller–Segel
• quasilinear parabolic system
• global existence

2010 MSC:

• 35K55
• 35K59
• 35Q92

Acknowledgements

The authors are very grateful to the anonymous reviewers for their carefully reading and valuable suggestions which greatly improved this work.

Disclosure statement

No potential conflict of interest was reported by the authors.