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Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions
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- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2609.06479
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Abstract
This paper studies the spectral properties and nonlinear asymptotic stability of a family of finite-time self-similar blow-up solutions to the three-dimensional Keller--Segel system constructed by matching interior and exterior profiles within the framework of matched asymptotic expansions. For every sufficiently large matching index $n$, the full linearized operator around the stationary state $U_n$ in self-similar variables is analyzed on $L^2(\mathbb R^3)$. Sturm zero counting in the radial mode, a wave operator that reduces the nonlocal $l=1$ equation to a local equation, and a Mellin--Newton quadratic form for all $l\ge2$ show that, after the scaling and translation modes and the finitely many genuinely unstable radial modes are removed, the remaining spectrum is separated from the imaginary axis by a positive distance. In addition, the optimal lower bound on the real parts of the spectrum is $1/4$ in every mode $l\ge2$. On the stable subspace, an exponentially decaying semigroup and a modified energy equivalent to the $L^2$ norm are constructed, and the logarithmic asymptotic decay rate of the semigroup norm is proved to equal the stable spectral gap. Finally, modulation equations, $H^2$ energy estimates, control of the scaling derivative, and Brouwer's no-retraction theorem yield nonradial nonlinear asymptotic stability of the corresponding self-similar blow-up solutions after the initial coefficients in the finitely many unstable radial directions have been chosen suitably.
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