A quasilinear parabolic–elliptic chemotaxis-growth system with nonlinear secretion. Issue 1 (2nd January 2020) - Zhao, Jie; Mu, Chunlai; Wang, Liangchen; Lin, Ke
ABSTRACT: This paper deals with the parabolic–elliptic chemotaxis-growth system with nonlinear secretion u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( χ u ∇ v ) + a u − b u r, x ∈ Ω, t > 0, 0 = Δ v − v + u k, x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n ( n ≥ 2 ) with smooth boundary, where χ > 0, a >0, b >0, r >1, k ≥ 1 and D ( u ) is a smooth function satisfying D ( u ) > c D u m − 1 with some c D > 0 and m ≥ 1 . It is shown, either r < k + 1, m > k + 1 − 2 n, or r > k + 1, or r = k + 1, b > b ∗, where b ∗ = 0, if m > k + 1 − 2 n, n ( k + 1 − m ) − 2 n ( k + 1 − m ), if m ≤ k + 1 − 2 n, then the system admits a unique global bounded classical solution. Moreover, we obtain the large time behavior of the solution for a specific logistic source. Finally, for the case D ( u ) ≡ 1 and r = k +1, large time behavior and convergence rate are established.