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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1811.09007
- arXiv:
- arXiv:1811.09007
- Bibcode:
- 2018arXiv181109007J
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- Discrete and Continuous Dynamical Systems,2020