On a fully parabolic chemotaxis system with nonlocal growth term

• Published in: Nonlinear analysis 2021-12, Vol.213, p.112518, Article 112518
• Main Authors: Negreanu, M., Tello, J.I., Vargas, A.M.
• Format: Article
• Language: English
• Subjects: Asymptotic behavior; Chemotaxis; Global existence of solutions; Nonlocal growth terms
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2021.112518

This article deals with a fully parabolic chemotaxis system describing the behavior of a biological species with density “u” which follows a chemical gradient with density “v”. The problem presents a nonlocal growth term defined by f(u)=ua0−a1uα+a2∫Ωuαdxand the system is given by the following two second order coupled parabolic equations ut−Δu=−div(χum∇v)+f(u),vt−Δv+v=uγ,in a bounded domain Ω with homogeneous Neumann boundary conditions and appropriate initial data. The parameters α, m, ai (i=1,2) and γ satisfy α≥1,m>1,γ≥1,α+1>m+γ,a1>0,a1−a2|Ω|>0.Under suitable assumptions on the initial data and the coefficients of the system, the global-in-time existence of classical solutions and the convergence to the steady state u∗=a01α(a1−a2|Ω|)1α,v∗=(u∗)γ,when a0>0, are proved in any space dimension.