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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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##### References:
 [1] X. Cao, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation,, Nonlinearity, 27, 1899, (2014) · Zbl 1301.35056 [2] L. Corrias, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions,, Milan J. Math, 72, 1, (2004) · Zbl 1115.35136 [3] L. Corrias, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces,, Mathematical and Computer Modelling, 47, 755, (2008) · Zbl 1134.92006 [4] M. A. Herrero, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore, 24, 633, (1997) · Zbl 0904.35037 [5] T. Hillen, A user’s guide to PDE models in a chemotaxis,, J. Math. Biology, 58, 183, (2009) · Zbl 1161.92003 [6] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jber. DMV, 105, 103, (2003) · Zbl 1071.35001 [7] D. Horstmann, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215, 52, (2005) · Zbl 1085.35065 [8] W. Jäger, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Tran. Amer. Math. Soc., 329, 819, (1992) · Zbl 0746.35002 [9] E. F. Keller, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26, 399, (1970) · Zbl 1170.92306 [10] O. A. Ladyžxenskaja, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society [11] T. Li, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms,, Math. Models Methods Appl. Sci. · Zbl 1322.35054 [12] T. Nagai, Blowup of Nonradial Solutions to Parabolic-Elliptic Systems Modeling Chemotaxis in Two-Dimensional Domains,, J. Inequal. Appl., 6, 37, (2001) · Zbl 0990.35024 [13] T. Nagai, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40, 411, (1997) · Zbl 0901.35104 [14] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl. 2, 2, (1997) · Zbl 0383.35057 [15] I. Tuval, Bacterial swimming and oxygen transport near contact lines,, Proc. Nat. Acad. Sci, 102, 2277, (2005) · Zbl 1277.35332 [16] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248, 2889, (2010) · Zbl 1190.92004 [17] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100, 748, (2013) · Zbl 1326.35053 [18] C. Xue, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70, 133, (2009) · Zbl 1184.35308
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