Stabilization in a chemotaxis-consumption model involving Robin-type boundary conditions

Abstract

The chemotaxis-consumption model{ut=∇⋅((u+1)α∇u)−∇⋅(u(u+1)β−1∇v),x∈Ω,t>00=Δv−uv,x∈Ω,t>0 is considered in a smooth bounded domain Ω⊂Rn under the boundary conditions (u+1)α∂νu=u(u+1)β−1∂νv and ∂νv=(γ−v)g on ∂Ω, with parameters α,β∈R, γ>0 and the nonnegative function g∈C1+ω(Ω¯) for some ω∈(0,1). If β−α≤1, it is proved that there exists a global bounded classical solution (u,v) for suitably regular initial data u0. Furthermore, for the given mass m=∫Ωu∞>0, the corresponding stationary system admits a unique classical solution (u∞,v∞), appearing as a large-time limit in the sense that if γ>0 is small enough, then‖u(⋅,t)−u∞‖L∞(Ω)+‖v(⋅,t)−v∞‖L∞(Ω)→0 as t→∞, with ∫Ωu∞=∫Ωu0=m>0.