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

In this paper, 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. \qquad \qquad (\star) \end{equation} is considered under Neumann boundary conditions in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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 ${\epsilon }_0>0$ such that for all suitably regular initial data $(u_0,v_0)$ satisfying $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^{\frac{n}{2}}(\mathrm{\Omega })}<{\epsilon }_0$ and $|\nabla v_0|_{L^{n}(\Omega)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and approaches the constant steady state $(m,m)$ with $m:=\frac{1}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\int }_{\mathrm{\Omega }}{u}_0$. 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.