Boundedness in a fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. (August 2022) - Chiyo, Yutaro; Yokota, Tomomi
Abstract: This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( G ( u ) χ ( v ) ∇ v ) + ∇ ⋅ ( H ( u ) ξ ( w ) ∇ w ), x ∈ Ω, t > 0, v t = d 1 Δ v + α u − β v, x ∈ Ω, t > 0, w t = d 2 Δ w + γ u − δ w, x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 1 ) is a bounded domain with smooth boundary, d 1, d 2, α, β, γ, δ > 0 are constants. Also, D, G, H ∈ C 2 ( [ 0, ∞ ) ) fulfill that a 0 ( s + 1 ) m − 1 ≤ D ( s ) ≤ a 1 ( s + 1 ) m − 1 with a 0, a 1 > 0 and m ∈ R ; G ( 0 ) = 0, 0 ≤ G ( s ) ≤ b 0 ( s + 1 ) q − 1 with b 0 > 0 and q < min 2, m + 1 ; H ( 0 ) = 0, 0 ≤ H ( s ) ≤ c 0 ( s + 1 ) r − 1 with c 0 > 0 and r < min 2, m + 1 , and χ, ξ satisfy that 0 ≤ χ ( s ) ≤ χ 0 s k 1 with χ 0 > 0 and k 1 > 1 ; 0 ≤ ξ ( s ) ≤ ξ 0 s k 2 with ξ 0 > 0 and k 2 > 1 . Global existence and boundedness in the case that w = 0 were proved by Ding (2018). However, there is no work on the above fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system.