Large time behavior of solutions for density-suppressed motility system in higher dimensions

Abstract

In this paper, we study the asymptotic stability of the following density-suppressed motility system(⁎){ut=Δ(γ(v)u)+μu(1−u),x∈Ω,t>0,vt=Δv+u−v,x∈Ω,t>0, in a bounded domain with smooth boundary, where the motility function γ(v) and μ are positive. Under the conditions that the motility function γ(v) has the lower-upper bounded and that μ is large enough, we derive that system (⁎) possesses a unique global solution in three-dimensional space. Moreover, we show that if the global classical solution exists of system (⁎) in any dimensional space, then the solution will converge to the equilibrium (1,1) exponentially as t→+∞ when μ>K016 with K0=max0≤v≤∞⁡|γ′(v)|2γ(v).