Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system

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• 作者：Hao Yu ^{yuhaomuu@126.com" class="auth_mail" title="E-mail the corresponding author} ; Qian Guo ^{424096826@qq.com" class="auth_mail" title="E-mail the corresponding author} ; Sining Zheng ; ^{snzheng@dlut.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Chemotaxis ; Attraction&ndash ; repulsion ; Blow-up ; Nonradial solutions ; Keller&ndash ; Segel system
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：34
• 期：Complete
• 页码：335-342
• 全文大小：600 K

文摘

This paper considers the attraction–repulsion chemotaxis system: u_t=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)$u_t=\Delta u-\chi \nabla \cdot \left(u\nabla v\right)+\xi \nabla \cdot \left(u\nabla w\right)$, 0=Δv+αu−βv$0=\Delta v+\alpha u-\beta v$, 0=Δw+γu−δw$0=\Delta w+\gamma u-\delta w$, subject to the non-flux boundary condition in a smooth bounded domain Ω⊂R²$\Omega \subset {\mathbb{R}}^2$, with χ,ξ≥0$\chi ,\xi \ge 0$, α,β,γ,δ>0$\alpha ,\beta ,\gamma ,\delta >0$. We establish the finite time blow-up conditions for nonradial solutions that the finite time blow-up occurs at x₀∈Ω$x_0\in \Omega$ whenever ∫_Ωu₀(x)dx>8π/(χα−ξγ)${\int }_{\Omega }{u}_0\left(x\right)dx>8\pi /(\chi \alpha -\xi \gamma )$ with χα−ξγ>0$\chi \alpha -\xi \gamma >0$, under ∫_Ωu₀(x)|x−x₀|²dx${\int }_{\Omega }{u}_0\left(x\right){|x-x_0|}^{2}dx$ sufficiently small. This does agree with the known blow-up conditions for radial solutions of the same model. The previous blow-up conditions for nonradial solutions are more complicated involving a classification to the sign of δ−β$\delta -\beta$.