吸引-排斥趋化系统解的全局存在性和有界性

Global existence and boundedness for an attraction-repulsion chemotaxis system

本文在光滑有界域??RN中讨论以下吸引-排斥趋化系统的齐次Neumann初边值问题:{u_(t)=Δu-x△↓·(u△↓)+ξ△↓·(u△↓w),x∈Ω,t>0,0=△u-βu+αu,x∈Ω,t>0,0=△w-δw+γu,x∈Ω,t>0,δu/δv=δu/δv=δw/δv=0,x∈δΩ,t>0,u(x,0)=u_(0)(x),x∈Ω,其中χ、ξ、β、α、δ和γ是正常数.当N=4时,在ξγ=χα和ξδλ_(0γ)fΩu_(0)<1/C_(GN)的假设条件下,本文提出一种新的方法来建立适当初值条件下上述系统全局经典解的存在性和有界性,这一成果为进一步理解和探索更高维空间下系统解的行为提供了重要的理论基础.

In this paper,we deal with the following attraction-repulsion chemotaxis system:{u_(t)=Δu-x△↓·(u△↓)+ξ△↓·(u△↓w),x∈Ω,t>0,0=△u-βu+αu,x∈Ω,t>0,0=△w-δw+γu,x∈Ω,t>0,δu/δv=δu/δv=δw/δv=0,x∈δΩ,t>0,u(x,0)=u_(0)(x),x∈Ω,under homogenous Neumann boundary conditions in a smoothly bounded domainΩ⊂R^(N),where,χ、ξ、β、α、δand γare positive constants.Many previous works have established the existence of global bounded classical solutions for the case N ≤3,but leave an open question for the case N≥4.In this paper,for the case N=4,we develop a new method to establish the existence and boundedness of global classical solutions for suitable initial data under the assumptionξγ=χαandξδλ_(0γ)fΩu_(0)<1/C_(GN),where C_(GN) andλ_(0) are some positive constants only depending onΩ.