Global existence and uniqueness of solutions to a chemotaxis system. Issue 16 (9th December 2020)

• Record Type: Journal Article
• Authors: Aïssa, Naïma; Balehouane, Abdelkhalek
• Is Part Of: Applicable analysis. Volume 99:Issue 16(2020)
• Page Start: 2833
• Page End: 2853
• Publication Date: 2020-12-09
• Subjects: Roger Temam; Nonlinear diffusion -- chemotaxis -- Keller–Segel -- quasilinear parabolic system -- global existence; 35K55 -- 35K59 -- 35Q92; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2019.1585530
• Languages: English
• ISSNs: 0003-6811

Abstract

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.