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

Abstract A fully parabolic chemotaxis system u t = Δ u − ∇ ⋅ ( u χ ( v ) ∇ v ) , v t = Δ v − 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 .