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

Abstract

In this paper, the fully parabolic Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=Δu-\nabla\cdot(u\nabla v), &(x,t)\in Ω\times (0,T),\\ v_t=Δv-v+u, &(x,t)\inΩ\times (0,T),\\ \end{array} \right. \qquad \qquad (\star) \end{equation} is considered under Neumann boundary conditions in a bounded domain $Ω\subset\mathbb{R}^n$ 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 $\|u_0\|_{L^{\frac{n}{2}}(Ω)}<\varepsilon_0$ and $\|\nabla v_0\|_{L^{n}(Ω)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and approaches the constant steady state $(m,m)$ with $m:=\frac{1}{|Ω|}\int_Ω u_0$. 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.