Boundedness, blowup and critical mass phenomenon in competing chemotaxis

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• 作者：Hai-Yang Jin^a ; ^{mahyjin@scut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Zhi-An Wang^b ; ^{mawza@polyu.edu.hk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35A01 ; 35B40 ; 35B44 ; 35K57 ; 35Q92 ; 92C17
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 January 2016
• 年：2016
• 卷：260
• 期：1
• 页码：162-196
• 全文大小：470 K

文摘

We consider the following attraction–repulsion Keller–Segel system: with homogeneous Neumann boundary conditions in a bounded domain 惟⊂R²$\mathrm{惟}\subset {\mathbb{R}}^2$ with smooth boundary. The system models the chemotactic interactions between one species (denoted by u) and two competing chemicals (denoted by v and w), which has important applications in Alzheimer's disease. Here all parameters 蠂, 尉, 伪, 尾, 纬 and 未   are positive. By constructing a Lyapunov functional, we establish the global existence of uniformly-in-time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction (i.e., 尉纬≥伪蠂$尉纬\ge 伪蠂$). If the attraction dominates (i.e., 尉纬<伪蠂$尉纬<伪蠂$), a critical mass phenomenon is found. Specifically speaking, we find a critical mass $m_{鈦?/mo>}=\frac{4蟺}{伪蠂-尉纬}$ such that the solution exists globally with uniform-in-time bound if M<m_鈦?/sub>$M<m_{鈦?/mo>}$ and blows up if cdf10193e2fb3947eadf1d37b37011" title="Click to view the MathML source">M>m_鈦?/sub>$M>m_{鈦?/mo>}$ and $M\notin \{\frac{4蟺m}{胃}:m\in {\mathbb{N}}^{+}\}$ where N⁺${\mathbb{N}}^{+}$ denotes the set of positive integers and M=∫_惟u₀dx$M={\int }_{\mathrm{惟}}{u}_{0}dx$ the initial cell mass.