(untitled) (Journal of Partial Differential Equations, 2025)
In this paper, we study the asymptotic behavior of solutions to a quasilinear two-species chemotaxis system with nonlinear sensitivity and nonlinear signal production under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R^n(n≥2) where the parameter μ_1,μ_2,γ_1,γ_2 are positive constants, τ ∈ {0,1}. The diffusion coefficients D_i,Si ∈ C^2([0,∞)) satisfy D_i(s)≥a_0(s+1)^−m_i, 0≤S_i(s)≤ b_0s(s+1)^β_i−1, s≥0, m_i,β_i ∈R, a_0,b_0 >0, i=1,2. Under the assumption of properly initial data regularity, we can find appropriate μ_i such that the globally bounded solution of this system satisfies the following relationship. (I) If a_1,a_2∈(0,1) and μ_1 and μ_2 are sufficiently large, then any globally bounded solution exponentially converges to (1-a_11-a_1a_2, (1-a_21-a_1a_2)^γ_1, 1-a_21-a_1a_2, (1-a_11-a_1a_2)^γ_2) as t→∞; (II) If a_1 >1>a_2 >0 and μ_2 is sufficiently large, then any globally bounded solution exponentially converges to (0,1,1,0) as t→∞; (III) If a_1=1>a_2 >0 and μ_2 is sufficiently large, then any globally bounded solution algebraically converges to (0,1,1,0) as t→∞.