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

We consider the initial-boundary value problem of a system of reaction-diffusion equations with density-dependent motility [Formula presented] 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 [Formula presented] if D>0 is suitably large.