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Sharp Long-Time Asymptotics for Chemotaxis with Free Boundary
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2021-04-13 , DOI: 10.1137/19m124678x
Hai-Liang Li , Benoít Perthame , Xinmei Wen

SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 2027-2083, January 2021.
The Patlak--Keller--Segel model can be used to model the nonlocal aggregation phenomena in the collective motion of cells or the evolution of the density of bacteria by chemotaxis. We consider the free boundary value problem for the Patlak--Keller--Segel model with homogeneous nonlinear degenerate diffusion in the present paper, which can be used to simulate the congested phenomena and the dynamical behaviors of cell motion with finite total mass and compactly supported density distribution. For the subcritical case, we prove that the cell density function exists globally in time and tends to the corresponding steady-state at an exponential time rate due to the balance between the nonlinear diffusion effect and nonlocal aggregation. For the supercritical case, we show that the global solution for the cell density exists and converges algebraically in time to the Barenblatt solution of the corresponding porous media equation due to the diffusion dominating mechanism. This provides a different viewpoint on the dynamical behaviors of the congested phenomena subject to the combined influences of nonlinear diffusion and nonlocal aggregation.


中文翻译:

具有自由边界的急性趋化性的长期尖锐渐近性

SIAM数学分析杂志,第53卷,第2期,第2027-2083页,2021年1月。
Patlak-Keller-Segel模型可用于模拟细胞集体运动中的非局部聚集现象,或通过趋化作用产生的细菌密度变化。我们考虑了具有均一非线性简并扩散的Patlak-Keller-Segel模型的自由边界值问题,该模型可用于模拟拥塞现象和有限总质量且受紧支撑的细胞运动的动力学行为密度分布。对于亚临界情况,由于非线性扩散效应和非局部聚集之间的平衡,我们证明了细胞密度函数在时间上全局存在,并趋于以指数时间速率处于相应的稳态。对于超临界情况,我们表明,由于扩散支配机制的存在,细胞密度的整体解存在并在时间上代数收敛到相应多孔介质方程的Barenblatt解。这对受非线性扩散和非局部聚集的综合影响的拥塞现象的动力学行为提供了不同的观点。
更新日期:2021-04-14
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