Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

summary:We study the chemotaxis system with singular sensitivity and logistic-type source: uₜ=Δ u-χ ∇ · (u ∇ v/ v) +ru-μ uᵏ, 0=Δ v-v+u under the non-flux boundary conditions in a smooth bounded domain Ω ⊂ Rⁿ, χ ,r,μ >0, $k>1$ and n≥ 1. It is shown with k∈ (1,2) that the system possesses a global generalized solution for n≥ 2 which is bounded when χ >0 is suitably small related to $r>0$ and the initial datum is properly small, and a global bounded classical solution for $n=1$.