On a fully parabolic chemotaxis system with nonlocal growth term. (December 2021) - Negreanu, M.; Tello, J.I.; Vargas, A.M.
Abstract: 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 ) = u a 0 − a 1 u α + a 2 ∫ Ω u α d x and the system is given by the following two second order coupled parabolic equations u t − Δ u = − div ( χ u m ∇ v ) + f ( u ), v t − Δ v + v = u γ, in a bounded domain Ω with homogeneous Neumann boundary conditions and appropriate initial data. The parameters α, m, a i ( i = 1, 2 ) and γ satisfy α ≥ 1, m > 1, γ ≥ 1, α + 1 > m + γ, a 1 > 0, a 1 − a 2 | 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 ∗ = a 0 1 α ( a 1 − a 2 | Ω | ) 1 α, v ∗ = ( u ∗ ) γ, when a 0 > 0, are proved in any space dimension.