Global solutions to a chemotaxis-growth system with signal-dependent motilities and signal consumption

This paper deals with an initial-boundary value problem about the chemotaxis-growth system(0.1){ut=∇(γ(v)∇u−uϕ(v)∇v)+μu(1−u),x∈Ω,t>0,vt=△v−uv,x∈Ω,t>0 in a bounded domain Ω⊂Rn(n≥2) with no-flux boundary conditions. Here one of the two density-dependent motility functions γ(v) describes the strength of diffusion while the other ϕ(v)=(α−1)γ′(v)(α>0) denotes the chemotactic sensitivity. It is proved that for a class generic motility functions there exists a unique global bounded classical solution to (0.1) with some suitable small initial data and some large μ. Furthermore, it asserts that the obtained global solution stabilizes to the spatially uniform equilibrium (1,0) in the sense that‖u(⋅,t)−1‖L∞(Ω)→0,‖v(⋅,t)‖W1,∞(Ω)→0ast→∞.