Global stability of Keller–Segel systems in critical Lebesgue spaces

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| $.