In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: u t = ∆u − χ∇·( u v ∇v)+ ru − µu k , 0 = ∆v − v + u under the non-flux boundary conditions in a smooth bounded convex domain Ω ⊂ R n , χ, r, µ > 0, k > 1 and n ≥ 2. It is shown that the system possesses a globally bounded classical solution if k > 3n−2 n , and r > χ 2 4 for 0 < χ ≤ 2, or r > χ − 1 for χ > 2. In addition, under the same condition for r, χ, the system admits a global generalized solution when k ∈ (2 − 1 n , 3n−2 n ], moreover this global generalized solution should be globally bounded provided r µ and the initial data u 0 suitably small.

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We consider a parabolic-parabolic Keller-Segel system of chemotaxis model with singular sensitivity u t = ∆u − χ∇ · ( u v ∇v), v t = k∆v − v + u under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 2), with χ, k > 0. It is proved that for any k > 0, the problem admits global classical solutions, wheneverThe global solutions are moreover globally bounded if n ≤ 8. This shows an exact way the size of the diffusion constant k of the chemicals v effects the behavior of solutions.

We consider a chemotaxis consumption system with singular sensitivity ut=normalΔu−χ∇·false(uvα∇vfalse), vt=εΔv−uv in a bounded domain normalΩ⊂Rn with χ,α,ε>0. The global existence of classical solutions is obtained with n=1. Moreover, for any global classical solution (u,v) to the case of n,α≥1, it is shown that v converges to 0 in the L∞‐norm as t→∞ with the decay rate established whenever ε∈(ε0,1) with ε0=maxfalse{0,1−χαfalse‖v0‖L∞false(normalΩfalse)α−1false}.