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Global stability of Keller–Segel systems in critical Lebesgue spaces

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  • This paper is concerned with the initial-boundary value problem for the classical Keller–Segel system

    in a bounded domain $ \Omega\subset\mathbb{R}^d $ with $ d\geq2 $ under homogeneous Neumann boundary conditions, where $ \gamma\geq0 $. We study the existence of non-trivial global classical solutions near the spatially homogeneous equilibria $ \rho = c\equiv\mathcal{M}>0 $ with $ \mathcal{M} $ being any given large constant which is an open problem proposed in [2,p. 1687]. More precisely, we prove that if $ 0<\mathcal{M}<1+\lambda_1 $ with $ \lambda_1 $ being the first positive eigenvalue of the Neumann Laplacian operator, one can find $ \varepsilon_0>0 $ such that for all suitable regular initial data $ (\rho_0,\gamma c_0) $ satisfying

    $\frac{1}{{|\Omega |}}\int_\Omega {{\rho _0}} dx - {\cal M} = \gamma \left( {\frac{1}{{|\Omega |}}\int_\Omega {{c_0}} dx - {\cal M}} \right) = 0\;\;\;\;\;\;\;\left( 2 \right)$


    ${\rho _0} - {\cal M}{_{{L^{d/2}}(\Omega )}} + \gamma \nabla {c_0}{_{{L^d}(\Omega )}} < {\varepsilon _0},\;\;\;\;\;\;\;\left( 3 \right)$

    problem (1) possesses a unique global classical solution which is bounded and converges to the trivial state $ (\mathcal{M},\mathcal{M}) $ exponentially as time goes to infinity. The key step of our proof lies in deriving certain delicate $ L^p-L^q $ decay estimates for the semigroup associated with the corresponding linearized system of (1) around the constant steady states. It is well-known that classical solution to system (1) may blow up in finite or infinite time when the conserved total mass $ m\triangleq\int_\Omega \rho_0 dx $ exceeds some threshold number if $ d = 2 $ or for arbitrarily small mass if $ d\geq3 $. In contrast, our results indicates that non-trivial classical solutions starting from initial data satisfying (2)-(3) with arbitrarily large total mass $ m $ exists globally provided that $ |\Omega| $ is large enough such that $ m<(1+\lambda_1)|\Omega| $.

    Mathematics Subject Classification: Primary: 35K51, 35K59; Secondary: 35Q92.


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  • [1] M. Ashbaugh and A. Levine, Inequalities for Dirichlet and Neumann eigenvalues of the laplacian for domains on sphere, Journées Équations Aux Dérivées Partielles, 1997, 1–15.
    [2] N. BellomoA. BelouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biology tissues, Math. Mod. Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.
    [3] X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dynam. Syst. Ser. A, 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.
    [4] T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Banach Center Publ., Polish Acad. Sci., Warsaw, 81 (2008), 105–117. doi: 10.4064/bc81-0-7.
    [5] K.J. Engel and R. Nagel, One-parameter Semigroup for Linear Evolution Equations, GTM194, Springer, 2000.
    [6] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.
    [7] D. Horstmann and G.-F. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Euro. J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.
    [8] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Different. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.
    [9] J. Jiang, Global stability of homogeneous steady states in scaling-invariant spaces for a Keller–Segel–Navier–Stokes system, J. Different. Equ., 267 (2019), 659-692.  doi: 10.1016/j.jde.2019.01.022.
    [10] J. Jiang, Eventual smoothness and exponential stabilization of global weak solutions to some chemotaxis systems, preprint, submitted.
    [11] J. JiangH. Wu and S. Zheng, Blow-up for a three dimensional Kelle–Segel model with consumption of chemoattractant, J. Different. Equ., 264 (2018), 5432-5464.  doi: 10.1016/j.jde.2018.01.004.
    [12] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.
    [13] A. Kiselev and X. Xu, Suppression of chemotactic explosion by mixing, Arch. Rational Mech. Anal., 222 (2016), 1077-1112.  doi: 10.1007/s00205-016-1017-8.
    [14] H. KozonoM. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller–Segel system coupled with the Navier–Stokes fluid, J. Funct. Anal., 270 (2016), 1663-1683.  doi: 10.1016/j.jfa.2015.10.016.
    [15] A. Lorz, A coupled Keller–Segel–Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.  doi: 10.4310/CMS.2012.v10.n2.a7.
    [16] T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156. 
    [17] T. NagaiT. Senba and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497.  doi: 10.32917/hmj/1206124609.
    [18] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Different. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.
    [19] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller–Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.
    [20] H. YuW. Wang and S. Zheng, Global classical solutions to the Keller–Segel–Navier–Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770.  doi: 10.1016/j.jmaa.2017.12.048.
    [21] S. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.
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