# American Institute of Mathematical Sciences

May  2015, 35(5): 1891-1904. doi: 10.3934/dcds.2015.35.1891

## Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

 1 School of Mathematical Science, Dalian University of Technology, Dalian 116023, China

Received  May 2014 Revised  September 2014 Published  December 2014

In this paper, the fully parabolic Keller-Segel system $$\label{problemAbstract}\left\{\begin{array}{ll} u_t=\Delta u-\nabla\cdot(u\nabla v), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ \end{array}\right.\tag{\star}$$ is considered under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^n$ with smooth boundary, where $n\ge 2$. We derive a smallness condition on the initial data in optimal Lebesgue spaces which ensure global boundedness and large time convergence. More precisely, we shall show that one can find $\varepsilon_0>0$ such that for all suitably regular initial data $(u_0,v_0)$ satisfying $\|u_0\|_{L^{\frac{n}{2}}(\Omega)}<\varepsilon_0$ and $\|\nabla v_0\|_{L^{n}(\Omega)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and converges to the constant steady state $(m,m)$ with $m:=\frac{1}{|\Omega|}\int_\Omega u_0$.
Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with ($\star$). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.
Citation: Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1891-1904. doi: 10.3934/dcds.2015.35.1891
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