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 76c8e623" title="Click to view the MathML source">Ω⊂Rⁿ$\mathrm{\Omega }\subset {\mathbb{R}}^n$ with 76480ec3dfa2c479bb257" title="Click to view the MathML source">n≥2$n\ge 2$ and 761f0cb01428cc7ced1fd90e">$f\in {\mathrm{L}}^{\infty }([0,\infty );{\mathrm{L}}^{\frac{n}{2}+{\delta }_0}(\mathrm{\Omega }))\cap C^{\alpha }(\mathrm{\Omega }\times (0,\infty ))$ with some α>0$\alpha >0$ and 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 n=2$n=2$ and f   being constant in time, requiring the nonnegative initial data u₀$u_0$ to fulfill the property ${\int }_{\mathrm{\Omega }}{u}_0\mathrm{d}x<4\pi$ 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 f≡0$f\equiv 0$ in the system above) to the case 6bb8672db71f42d8fc95b701f3f62" 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 76480ec3dfa2c479bb257" 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.