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February  2016, 36(2): 861-875. doi: 10.3934/dcds.2016.36.861

## Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems

 1 Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419 2 Department of Mathematics and Information Technology, The Hong Kong Institute of Education, Rm 19A, 1/F, Block D4, 10, Lo Ping Road, Tai Po, New Territories, Hong Kong, China

Received  February 2014 Revised  February 2015 Published  August 2015

We prove the global-in-time existence of intermediate weak solutions of the equations of chemotaxis system in a bounded domain of $\mathbb{R}^2$ or $\mathbb{R}^3$ with initial chemical concentration small in $H^1$. No smallness assumption is imposed on the initial cell density which is in $L^2$. We first show that when the initial chemical concentration $c_0$ is small only in $H^1$ and $(n_0-n_\infty,c_0)$ is smooth, the classical solution exists for all time. Then we construct weak solutions as limits of smooth solutions corresponding to mollified initial data. Finally we determine the asymptotic behavior of the global solutions.
Citation: Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861
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