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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.

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