Large mass boundary condensation patterns in the stationary Keller-Segel system

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• 作者：Manuel del Pino^a ; ^{delpino@dim.uchile.cl" class="auth_mail" title="E-mail the corresponding author} ; Angela Pistoia^b ; ^{angela.pistoia@uniroma1.it" class="auth_mail" title="E-mail the corresponding author} ; Giusi Vaira^b ; ^{vaira.giusi@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35J60 ; secondary ; 35B33 ; 35J20
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 September 2016
• 年：2016
• 卷：261
• 期：6
• 页码：3414-3462
• 全文大小：507 K

文摘

We consider the boundary value problem where Ω is a bounded smooth domain in R²${\mathbb{R}}^2$, λ>0$\lambda >0$ and ν is the inner normal derivative at ∂Ω. This problem is equivalent to the stationary Keller–Segel system from chemotaxis.

We establish the existence of a solution u_λ$u_{\lambda }$ which exhibits a sharp boundary layer along the entire boundary ∂Ω as λ→0$\lambda \to 0$. These solutions have large mass in the sense that ∫_Ωλe^u_λ∼|log⁡λ|${\int }_{\mathrm{\Omega }}\lambda e^{u_{\lambda }}\sim |\log \lambda |$.