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This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics (0.1) { u t = Δ u − χ 1 ∇ ⋅ (u w ∇ w) + μ 1 u (1 − u − a 1 v) , t > 0 , x ∈ Ω , v t = Δ v − χ 2 ∇ ⋅ (v w ∇ w) + μ 2 v (1 − v − a 2 u) , t > 0 , x ∈ Ω , w t = Δ w − w + u + v , t > 0 , x ∈ Ω , ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = 0 , t > 0 , x ∈ ∂ Ω , u (0 , x) = u 0 (x) , v (0 , x) = v 0 (x) , w (0 , x) = w 0 (x) , x ∈ Ω , where Ω ⊂ R N (N ≥ 2) is a bounded smooth convex domain, and the parameters χ 1 , χ 2 , μ 1 , μ 2 , a 1 and a 2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ 1 , χ 2 ∈ (0 , 1 2) for N = 2 , 3 or χ 1 , χ 2 ∈ (0 , N − 2 N − 1) for N ≥ 4. Moreover, if min ⁡ { μ 1 , μ 2 } > max ⁡ { χ 1 , χ 2 } 2 4 , we obtain the uniformly lower bound for w. Finally, when χ 1 , χ 2 are suitably small, it is proved that if 0 < a 1 , a 2 < 1 , then the solution (u , v , w) converges to (1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , 2 − a 1 − a 2 1 − a 1 a 2 ) in L ∞ norm as t → ∞ ; if 0 < a 2 < 1 ≤ a 1 , then the solution (u , v , w) converges to (0 , 1 , 1) in L ∞ norm as t → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasilinear chemotaxis model with singular sensitivity and indirect signal production (0.1) u t = ∇ ⋅ (D (u) ∇ u) − χ ∇ ⋅ (u v k ∇ v) + u (a − b u) , t > 0 , x ∈ Ω , v t = Δ v − v + w , t > 0 , x ∈ Ω , w t = Δ w − w + u , t > 0 , x ∈ Ω , under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω ⊂ R N , N ≥ 3 , where, χ , a , b > 0 , k ∈ (0 , 1 2) ∪ (1 2 , 1 ] and D (u) is smooth function satisfying D (u) ≥ (u + 1) α with α > 0. By constructing suitable Lyapunov functional, we prove that if a > χ 2 4 , there exists a positive constant m ∗ such that ∫ Ω u ≥ m ∗. Based on above result, the lower bound estimates for w and v have been established. When b is large enough and α > max { 0 , 1 − 4 N } , we further prove that the system has a global bounded classical solution. Finally, it is shown that the solution exponentially converges to the constant stationary solution (a b , a b , a b) as t → ∞. [ABSTRACT FROM AUTHOR]