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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:

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