Global well-posedness and asymptotic stabilization for chemotaxis system with signal-dependent sensitivity

A fully parabolic chemotaxis system u _t =Δu−∇⋅(uχ(v)∇v),v _t =Δv−v+u, in a smooth bounded domain Ω⊂R ^N , N≥2 with homogeneous Neumann boundary conditions is considered, where the non-negative chemotactic sensitivity function χ satisfies χ(v)≤μ(a+v) ^−k , for some a≥0 and k≥1. It is shown that a novel type of weight function can be applied to a weighted energy estimate for k>1. Consequently, the range of μ for the global existence and uniform boundedness of classical solutions established by Mizukami and Yokota [23] is enlarged. Moreover, under a convexity assumption on Ω, an asymptotic Lyapunov functional is obtained and used to establish the asymptotic stability of spatially homogeneous equilibrium solutions for k≥1 under a smallness assumption on μ. In particular, when χ(v)=μ/v and N<8, it is shown that the spatially homogeneous steady state is a global attractor whenever μ≤1/2.