Abstract
This paper deals with an initial-boundary value problem for the system
which has been proposed as a model for the spatio-temporal evolution of populations of swimming aerobic bacteria. It is known that in bounded convex domains \({\Omega \subset \mathbb{R}^2}\) and under appropriate assumptions on the parameter functions χ, f and ϕ, for each \({\kappa\in\mathbb{R}}\) and all sufficiently smooth initial data this problem possesses a unique global-in-time classical solution. The present work asserts that this solution stabilizes to the spatially uniform equilibrium \({(\overline{n_0},0,0)}\) , where \({\overline{n_0}:=\frac{1}{|\Omega|} \int_\Omega n(x,0)\,{\rm d}x}\) , in the sense that as t→∞,
hold with respect to the norm in \({L^\infty(\Omega)}\) .
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Winkler, M. Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Arch Rational Mech Anal 211, 455–487 (2014). https://doi.org/10.1007/s00205-013-0678-9
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DOI: https://doi.org/10.1007/s00205-013-0678-9