Global Stability of Keller--Segel Systems in Critical Lebesgue Spaces

Abstract

In this paper, we study the global stability of classical solutions to a Keller--Segel equations in scaling-invariant spaces. We prove that for any given $0<\mathcal{M}<1+{\lambda }_1$ with ${\lambda }_1$ being the first eigenvalue of Neumann Laplacian, the initial--boundary value problem of the Keller--Segel system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to $(\mathcal{M},\mathcal{M})$ in the norm of $L^{d/2}(\Omega )\times {\dot{W}}^{1,d}(\Omega )$ and satisfies a natral average mass condition. Our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup. Our result suggests a new observation that nontrivial classical solution for Keller--Segel equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large, correspondingly.

Cited by in corpus (2)

• Asymptotically almost Periodic Solutions of the parabolic-parabolic Keller-Segel systems on bounded domains
• On pseudo almost Periodic Solutions of the parabolic-elliptic Keller-Segel systems