We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge to the prescribed constant equilibrium states as time goes to infinity. Then we prove that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model when the chemical diffusion coefficient tends to zero.

中文翻译：

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多维趋化性平衡定律系统的大时间行为和扩散极限

我们考虑从化学趋化模型导出的具有多个空间维奇异敏感性的平衡律系统的柯西问题。利用能量方法，当只有初始数据的一阶空间导数的能量足够小时，并且证明解收敛于规定的恒定平衡态时，我们首先证明柯西问题的经典解的全局适定性。随着时间的流逝到无穷大。然后我们证明当化学扩散系数趋于零时，完全耗散模型的解收敛到相应的部分耗散模型的解。