Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production.

In this paper we study the asymptotic behaviors of global solutions to the fully parabolic chemotaxis system: u t = ∇ ⋅ (D (u) ∇ u − S (u) ∇ v) + r u − μ u 1 + σ , v t = Δ v − v + u γ , subject to the homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R n (n ≥ 2), where parameters μ , σ , γ > 0 , r ∈ R , and the nonlinearity D , S ∈ C 2 ([ 0 , ∞)) are supposed to generalize the prototypes D (u) ≥ a 0 (u + 1) − α , 0 ≤ S (u) ≤ b 0 u (u + 1) β − 1 with a 0 , b 0 > 0 and α , β ∈ R. We first consider the case of r > 0 and provide a boundedness result under α + β + γ < 2 n , or β + γ < 1 + σ , or β + γ = 1 + σ with large μ > 0. The main result is concerned with the asymptotic stability when damping effects of logistic source are strong enough. Specifically, there is μ 0 > 0 independent of initial data, such that the bounded classical solution (u , v) satisfies (u , v) → ((r μ) 1 σ , (r μ) γ σ ) in L ∞ (Ω) exponentially under conditions of μ > μ 0 and r > 0. For the case of r < 0 , the trivial constant equilibria in the model is obtained in a priori way, that is, any bounded solution (u , v) satisfies (u , v) → (0 , 0) in L ∞ (Ω) exponentially, regardless of the size of μ > 0. [ABSTRACT FROM AUTHOR]