Global stability of Keller–Segel systems in critical Lebesgue spaces

Jie Jiang

doi:10.3934/dcds.2020025

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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

Jie Jiang ^,

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| $

Keywords: Chemotaxis, Keller–Segel model, global solutions, global stability.

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

Citation: Jie Jiang. Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 609-634. doi: 10.3934/dcds.2020025