发表机构
College of Computing, Georgia Institute of Technology; Department of Mathematics and Systems Engineering, Florida Institute of Technology（佐治亚理工学院计算学院; 佛罗里达理工学院数学与系统工程系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究抛物型方程解算子对延迟测度的依赖性，通过加权谱和与实现恒等式，在Dirichlet拉普拉斯情形下得到尖锐的最坏情形稳定性模，并给出谱准则与半线性推广。
AI 中文摘要
我们研究了抛物型解算子对描述延迟律的有限符号测度的依赖性。对于具有紧逆的正自伴生成元，一个加权的二进谱和刻画了半群积分记忆扰动的范数。一个精确的实现恒等式将匹配的下界估计转移到固定线性方程中的两个正点延迟，使用一个共同的光滑有限谱历史。对于非空有界开集上的Dirichlet拉普拉斯算子，在连续$L^2$值历史的有界球上，尖锐的最坏情形模为$d\sqrt{\log(e/d)}$，其中$d$是延迟测度之间的有界Lipschitz距离。这是一个粗糙历史端点结果：阶$\gamma>1/2$的对数空间正则性在线性模型中恢复了分辨率一致的Lipschitz稳定性。临界阶$\gamma=1/2$保留了平方根双对数损失。更一般地，在占据谱带上的倒数平方可和性准则给出了精确的加权阈值，包括稀疏谱。结果包括尖锐的有限分辨率Lipschitz常数、记忆测度指定中点求积的最坏情形误差，以及在Hilbert空间值局部Lipschitz假设下的半线性上界估计。这些结果区分了对延迟律的敏感性与空间逼近误差，并且不排除在粗糙历史球上对正时间状态的均匀逼近。
英文摘要
We study the dependence of parabolic solution operators on a finite signed measure describing the delay law. For a positive self-adjoint generator with compact inverse, a weighted dyadic spectral sum characterizes the norm of the semigroup-integrated memory perturbation. An exact realization identity transfers the matching lower estimate to two positive point delays in a fixed linear equation, using a common smooth, finite-spectral history. For the Dirichlet Laplacian on a nonempty bounded open set, the sharp worst-case modulus on a bounded ball of continuous $L^2$-valued histories is $d\sqrt{\log(e/d)}$, where $d$ is the bounded-Lipschitz distance between the delay measures. This is a rough-history endpoint result: logarithmic spatial regularity of order $γ>1/2$ restores resolution-uniform Lipschitz stability in the linear model. The critical order $γ=1/2$ retains a square-root double-logarithmic loss. More generally, a reciprocal-square summability criterion over occupied spectral bands gives the exact weighted threshold, including sparse spectra. Consequences include sharp finite-resolution Lipschitz constants, worst-case errors for prescribed midpoint quadrature of the memory measure, and semilinear upper estimates under Hilbert-space-valued local Lipschitz assumptions. The results distinguish sensitivity to the delay law from spatial approximation error and do not preclude uniform approximation of positive-time states on rough history balls.
Comments25 pages
