摘要
本文在光滑有界域Ω⊂RN中讨论以下吸引-排斥趋化系统的齐次Neumann初边值问题:{ut=△u-x▽·(u▽v)+ξ▽·(u▽w),x ∈Ω,t＞0,0=△v-βv+αu,x∈Ω,t＞0,0=△w-δw+γu,x∈Ω,t＞0,∂u/∂v=∂v/∂v=∂w/∂v=0,x∈∂Ω,t＞0,u(x,0)=u0(x),x∈Ω,其中x、ξ、β、α、δ和γ是正常数.当N=4时,在ξγ=xα和ξδλ0γ∫Ωu0＜1/CGN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.
Abstract
In this paper,we deal with the following attraction-repulsion chemotaxis system:{ut=△u-x▽·(u▽v)+ξV·(u▽w),x ∈ Ω,t＞0,0=△v-βv+αu,x ∈ Ω,t＞0,0=△w-δw+γu,x ∈ Ω,t＞0,∂u/∂v=∂v/∂v=∂w/∂v=0,x∈∂Ω,t＞0,u(x,0)=u0(x),x ∈ Ω,under homogenous Neumann boundary conditions in a smoothly bounded domain Ω C RN,where x,ξ,β,α,δ and γ are 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 assumption ξγ=xα and ξδλ0γ ∫Ω u0＜1/CGN,where CGN and λ0 are some positive constants only depending on Q.
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