Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions
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 Un in self-similar variables is analyzed on L2(ℝ3). 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≥2 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≥2. On the stable subspace, an exponentially decaying semigroup and a modified energy equivalent to the L2 norm are constructed, and the logarithmic asymptotic decay rate of the semigroup norm is proved to equal the stable spectral gap. Finally, modulation equations, H2 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.
Related
- One-dimensional nodal solutions of the Lane--Emden problem in cylinders: asymptotic behaviour and energy stability
- Infinitely many nonradial positive solutions for two-coupled nonlinear Schrödinger equations in bbrN without symmetry
- Radial and nonradial solutions for nonlinear scalar field equations and a new proof for certain Gagliardo-Nirenberg type inequality on the plane
- Asymptotic stability and sharp decay near homogeneous solutions of a nonlinearly damped Schrödinger equation on the torus
- Asymptotic pseudo-periodicity of solutions for nonlinear fractional evolution equations
BibTeX
@misc{li2026optimal,
title = {{Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions}},
author = {Te Li and Yuwei Sun and Kaiqiang Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06479},
eprint = {2609.06479},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06479},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions
AU - Te Li
AU - Yuwei Sun
AU - Kaiqiang Zhang
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06479
UR - https://arxiv.org/abs/2609.06479
AB - 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(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≥2 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≥2. 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.
ER -