一类带有非线性产生的吸引-排斥趋化系统解的整体有界性

Global Boundedness in an Nonlinear Attraction-Repulsion Chemotaxis System with Nonlinear Productions

讨论了一类带有非线性产生和logistic源的吸引-排斥趋化系统,ut=▽·(D(u)▽u)-▽·(Φ(u)▽v)+▽·(Ψ(u)▽w)+f(u),0=Δv+αuk-βv,0=Δw+γul-δw,在有界域Ω■Rn(n≥1)上并给定齐次Neumann边界条件,非负函数D,Φ,Ψ∈C2[0,+∞)满足D(s)≥(s+1)p(s≥0),Φ(s)≤χsq,Ψ(s)≥ξsg(s>1),且f(s)≤s(a-bsd)(s>0),f(0)≥0.证明了,如果吸引机制被排斥机制,logistic源及细胞自身的非线性扩散机制三者之一占优,即q+k<max {g+l,d+1,2/n+p+1},解将会是整体有界的.此外,在q+k=g+l=d+1,q+k=g+l>d+1或q+k=d+1>g+l这三种平衡条件下,解的有界性依赖于相关系数的大小.这扩展了 Tian,He和Zheng对拟线性吸引-排斥趋化系统关于解的有界性的结论.

In this paper,we study the attraction-repulsion chemotaxis system with nonlinear productions and logistic source,ut=▽·(D(u)▽u)-▽·(Φ(u)▽v)+▽·(Ψ(u)▽w)+f(u),0=Δv+αuk-βv,0=Δw+γul-δw,in bounded domain Ω?Rn(n≥1),subject to the homogeneous Neumann boundary conditions,D,Φ,Ψ ∈C2 [0,+∞) nonnegative,with D(s)≥(s+1)p for s≥0,Φ(s)≤χsq,Ψ(s)≥ξsg for s> 1,and f satisfying f(s) ≤ s(a-bsd) for s> 0,f(0)≥0.It is proved that if the attraction is dominated by one of the repulsion,logistic scource and the nonliner diffusion mechanisms with q+k d+1 or q+k=d+1> g+l,the boundedness of solutions would be determined by the sizes of the coefficients involved.This extends the global boundedness criteria established by Tian,He and Zheng for the quasilinear attraction-repulsion chemotaxis system.