Global solvability for an indirect consumption chemotaxis system with signal-dependent motility

This paper considers the indirect signal consumption-chemotaxis system with signal-dependent motility in a smooth bounded domain Ω⊂Rⁿ(n≤3), as given by u_t=Δ(ϕ(v)u),v_t=Δv−vw,w_t=dΔw−w+u, where the motility function ϕ∈C³((0,∞)),ϕ>0 on (0,∞), which generalizes ϕ(v)=v^α,α∈R. Based on point-wise positive lower bound estimate of v, it is shown that for any suitably regular initial data, the corresponding initial–boundary value problem admits global smooth solutions.