Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility

Abstract

We consider the initial-boundary value problem of a system of reaction-diffusion equations with density-dependent motility(⁎){ut=Δ(γ(v)u)+αuF(w)−θu,x∈Ω,t>0,vt=DΔv+u−v,x∈Ω,t>0,wt=Δw−uF(w),x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,(u,v,w)(x,0)=(u0,v0,w0)(x),x∈Ω, in a bounded domain Ω⊂R2 with smooth boundary, α and θ are non-negative constants and ν denotes the outward normal vector of ∂Ω. The random motility function γ(v) and functional response function F(w) satisfy the following assumptions:•γ(v)∈C3([0,∞)),0<γ1≤γ(v)≤γ2,|γ′(v)|≤η for all v≥0;•F(w)∈C1([0,∞)),F(0)=0,F(w)>0in(0,∞)andF′(w)>0on[0,∞) for some positive constants γ1,γ2 and η. Based on the method of energy estimates and Moser iteration, we prove that the problem (⁎) has a unique classical global solution uniformly bounded in time. Furthermore we show that if θ>0, the solution (u,v,w) will converge to (0,0,w⁎) in L∞ with some w⁎>0 as time tends to infinity, while if θ=0, the solution (u,v,w) will asymptotically converge to (u⁎,u⁎,0) in L∞ with u⁎=1|Ω|(‖u0‖L1+α‖w0‖L1) if D>0 is suitably large.