一族三维Keller--Segel自相似爆破解的最优谱下界与非径向非线性渐近稳定性
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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中文总结 AI 辅助
本文通过谱分析和能量方法，证明了三维Keller--Segel系统一族自相似爆破解在去除不稳定模式后的非径向非线性渐近稳定性，并给出了最优谱下界1/4。
中文摘要 AI 辅助
本文研究三维Keller--Segel系统中一族有限时间自相似爆破解的谱性质和非线性渐近稳定性，这些解是在匹配渐近展开框架内通过匹配内部和外部轮廓构造的。对于每个足够大的匹配指标$n$，在自相似变量中围绕稳态$U_n$的完整线性化算子被在$L^2(\mathbb R^3)$上分析。径向模式中的Sturm零点计数、将非局部$l=1$方程约化为局部方程的波算子，以及适用于所有$l\ge2$的Mellin--Newton二次型表明，在去除标度和平移模式以及有限多个真正不稳定的径向模式后，剩余谱与虚轴的距离为正。此外，在每个模式$l\ge2$中，谱实部的最优下界为$1/4$。在稳定子空间上，构造了指数衰减半群和与$L^2$范数等价的修正能量，并证明了半群范数的对数渐近衰减率等于稳定谱隙。最后，调制方程、$H^2$能量估计、标度导数的控制以及Brouwer无收缩定理，在有限多个不稳定径向方向上的初始系数被适当选择后，给出了相应自相似爆破解的非径向非线性渐近稳定性。
英文摘要
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.
