Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities

We consider the Keller-Segel-type migration-consumption system involving signal-dependent motilities, {ut=Δ(uϕ(v)),vt=Δv−uv,in smoothly bounded domains Ω⊂Rn, n≥1. Under assumption that ϕ∈C1([0,∞)) is positive on [0,∞), and for nonnegative initial data from (C0(Ω¯))⋆×L∞(Ω), previous literature has provided results global existence of certain very weak solutions with possibly quite poor regularity properties, large time stabilization toward semitrivial equilibria respect to topology in (W1,2(Ω))⋆×L∞(Ω).The present study reveals fact enjoy significantly stronger features when 0<ϕ∈C3([0,∞)) belong (W1,∞(Ω))2: It firstly shown, namely, then case n≤2 an associated no-flux initial-boundary value problem even admits a classical solution, each these stabilizes sense as t→∞ we have u(⋅,t)→1|Ω|∫Ωu0andv(⋅,t)→0(⋆)even norm L∞(Ω) both components.In n≥3, secondly, some genuine are found exist globally, inter alia satisfying ∇u∈Lloc4/3(Ω¯×[0,∞);Rn). In particular three-dimensional setting, any such solution seen become eventually smooth satisfy (★).