Global dynamics for a chemotaxis consumption system with signal-dependent motility and logistic source

Abstract

This paper deals with the chemotaxis consumption model{ut=Δ(uϕ(v))+au−buγ,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥1). Here the parameters a>0, b>0 and γ>1, and the motility function ϕ satisfies ϕ∈C3([0,∞)) with ϕ(s)>0 for all s≥0. For all suitably regular initial data, if one of the following cases holds: (i) n≤2, γ>1; (ii) n≥3, γ>2; (iii) n≥3, γ=2 and b is large, then this system possesses global bounded classical solutions, while if n≥3 and γ∈(1,2], then this system admits at least one global weak solution, which becomes smooth after some waiting time and which will converge to the constant equilibrium ((ab)1γ−1,0).