| Contents |
| The fully parabolic Keller-Segel system
\begin{equation}
\left\{
\begin{array}{llc}
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.
\end{equation}
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. |
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