Boundedness in a Keller-Segel system with external signal production

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• 作者：Tobias Black ^{tblack@math.upb.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Chemotaxis ; Boundedness ; External signal production ; Critical mass
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：436-455
• 全文大小：501 K

文摘

We study the Neumann initial-boundary problem for the chemotaxis system in a smooth, bounded domain b8905626058a3e0a20fbd876c8e623" title="Click to view the MathML source">Ω⊂Rⁿ$\mathrm{\Omega }\subset {\mathbb{R}}^n$ with 80ec3dfa2c479bb257" title="Click to view the MathML source">n≥2$n\ge 2$ and $f\in {\mathrm{L}}^{\infty }([0,\infty );{\mathrm{L}}^{\frac{n}{2}+{\delta }_0}(\mathrm{\Omega }))\cap C^{\alpha }(\mathrm{\Omega }\times (0,\infty ))$ with some e91c58cb1c670c" title="Click to view the MathML source">α>0$\alpha >0$ and 971026" title="Click to view the MathML source">δ₀∈(0,1)${\delta }_0\in (0,1)$. First we prove local existence of classical solutions for reasonably regular initial values. Afterwards we show that in the case of e4ce9bad4c0dac797db03b1a3b1680d" title="Click to view the MathML source">n=2$n=2$ and f   being constant in time, requiring the nonnegative initial data e9b188" title="Click to view the MathML source">u₀$u_0$ to fulfill the property 805f9218a30f75a2">${\int }_{\mathrm{\Omega }}{u}_0\mathrm{d}x<4\pi$ ensures that the solution is global and remains bounded uniformly in time. Thereby we extend the well known critical mass result by Nagai, Senba and Yoshida for the classical Keller–Segel model (coinciding with b8444e7f11a85d934ba18" title="Click to view the MathML source">f≡0$f\equiv 0$ in the system above) to the case b8672db71f42d8fc95b701f3f62" title="Click to view the MathML source">f≢0$f\not\equiv 0$. Under certain smallness conditions imposed on the initial data and f   we furthermore show that for more general space dimension 80ec3dfa2c479bb257" title="Click to view the MathML source">n≥2$n\ge 2$ and f not necessarily constant in time, the solutions are also global and remain bounded uniformly in time. Accordingly we extend a known result given by Winkler for the classical Keller–Segel system to the present situation.