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January  2020, 40(1): 609-634. doi: 10.3934/dcds.2020025

Global stability of Keller–Segel systems in critical Lebesgue spaces

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, HuBei Province, China

* Corresponding author

Received  June 2019 Revised  July 2019 Published  October 2019

This paper is concerned with the initial-boundary value problem for the classical Keller–Segel system
$\left\{ {\begin{array}{*{20}{l}}{{\rho _t} - \Delta \rho = - \nabla \cdot (\rho \nabla c),\qquad }&{x \in \Omega ,\;t}&{ > 0}\\{\gamma {c_t} - \Delta c + c = \rho ,\qquad }&{x \in \Omega ,\;t}&{ > 0}\end{array}} \right.\;\;\;\;\;\;\left( 1 \right)$
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
$\frac{1}{{|\Omega |}}\int_\Omega {{\rho _0}} dx - {\cal M} = \gamma \left( {\frac{1}{{|\Omega |}}\int_\Omega {{c_0}} dx - {\cal M}} \right) = 0\;\;\;\;\;\;\;\left( 2 \right)$
and
${\rho _0} - {\cal M}{_{{L^{d/2}}(\Omega )}} + \gamma \nabla {c_0}{_{{L^d}(\Omega )}} < {\varepsilon _0},\;\;\;\;\;\;\;\left( 3 \right)$
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| $
.
Citation: Jie Jiang. Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 609-634. doi: 10.3934/dcds.2020025
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K.J. Engel and R. Nagel, One-parameter Semigroup for Linear Evolution Equations, GTM194, Springer, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Jiang, Eventual smoothness and exponential stabilization of global weak solutions to some chemotaxis systems, preprint, submitted. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K.J. Engel and R. Nagel, One-parameter Semigroup for Linear Evolution Equations, GTM194, Springer, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Jiang, Eventual smoothness and exponential stabilization of global weak solutions to some chemotaxis systems, preprint, submitted. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

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