Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension

Abstract

This manuscript considers the no-flux initial-boundary problem for the migration-consumption system{ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) in a smoothly bounded domain Ω⊂Rn, n≥1, where ϕ suitably generalizes the singular prototype given byϕ(ξ)=ξ−α,ξ>0, with α>0.It is firstly shown that for such diffusion singularities of arbitrary strength, and for any given initial data u0 and v0 from W1,∞(Ω) with u0≥0 and v0>0 in Ω‾, a so-called very weak-strong solution (u,v) with (u,v)|t=0=(u0,v0) can be constructed. Under the additional restrictions that 2≤n≤5 and α>n−26−n, and under an additional assumption on ϕ′, it is furthermore asserted that the flux ∇(uϕ(v)) belongs to some reflexive Lebesgue space, and that (⋆) is satisfied in a standard weak sense.Finally, in the one-dimensional case a statement on global classical solvability is derived under the mere condition that ϕ∈C3((0,∞)) be positive.