一类Keller-Segel趋化模型解在高维空间R^(N)(N≥3)的爆破问题

Blow-up Problems of a Class of Keller-Segel Chemotaxis Models in High-dimensional Space R^(N)(N≥3)

KellerSegel体系在数学生物学、理论物理和工程学等方面都具有广泛应用,是应用数学领域的研究热点问题之一.考虑宏观的非线性Keller Segel趋化模型,利用能量方法,首先构造一个能量表达式,然后运用高维Soblev嵌入不等式和一些微分不等式技巧,推导出能量所满足的一阶微分不等式,最终通过求解该不等式,得到KellerSegel趋化模型爆破时间的下界.将以往的结果由低维空间推广到高维空间.

The Keller-Segel system,which has a wide range of applications in mathematical biology,engineering,and theoretical physics,is one of the frontiers of the research in the field of applied mathematics.We consider a macroscopic nonlinear Keller-Segel convergence model and use energy methods in this paper.First,we create an expression of energy.Then we use the high-dimensional Sobolev embedding inequality and some basic differential inequality techniques to derive the first-order differential inequality that the energy satisfies.Finally,we obtain a lower bound on the outbreak time of the Keller-Segel convergence model by solving this inequality.This article can generalize the previous results from low-dimensional space to high-dimensional space.