Global solutions to a chemotaxis system with singular density-suppressed motility and superlinear consumption

Abstract

In this paper, we consider a migration-consumption taxis system involving singular density-suppressed motility and superlinear consumption(⋆){ut=Δ(uϕ(v)),vt=Δv−uvm in a bounded smooth domain Ω⊂Rn(n≥2) with no-flux boundary conditions. Here, m≥1 is a constant. The motility function ϕ generalizes the singular prototype given by ϕ(ξ)=ξ−α(ξ>0) with α>0. In Tao and Winkler (2023) [32], Tao and Winkler studied the system (⋆) with m=1 and obtained respectively the global existence of very weak-strong and weak-strong solutions. We study the system (⋆) with m≥1 and prove that a so-called very weak-strong solution can be established for arbitrarily large initial data in W1,∞(Ω). Moreover, under the additional constraints of(⋆⋆)(6−n)α+2m(n+2)−(3n+2)>0, and with an additional assumption on ϕ′, the model (⋆) possesses a global weak-strong solution. In particular, when 2≤n≤5 and m=1, (⋆⋆) is equivalent to α>n−26−n, which consistent with the result of Tao and Winkler. However, our result contains the case n≥6. When n=6, (⋆⋆) is equivalent to m>54 without any requirement for α>0. When n>6, (⋆⋆) is equivalent to m>3n+22n+4 and 0<α<2m(n+2)−(3n+2)n−6.