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Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. (English) Zbl 06384058
Summary: In this paper, the fully parabolic Keller-Segel system \[ \begin{cases} u_t=\Delta u-\nabla \cdot(u\nabla v),\quad &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, \quad &(x,t)\in\Omega\times (0,T), \end{cases} \tag{\(\star\)} \] is considered under Neumann boundary conditions in a bounded domain \(\Omega \subset \mathbb {R}^n\) with smooth boundary, where \(n\geq 2\). 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 \(\|u_0\|_{L^{\frac{n}{2}}(\Omega)}<\epsilon_0\) and \(\|\nabla v_0\|_{L^{n}(\Omega)}<\epsilon_0\), 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.

MSC:
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
92C17 Cell movement (chemotaxis, etc.)
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