发表机构
Team FEM（FEM团队）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究退化情形下正型记忆扩散方程，探讨完全单调记忆项能否替代缺失强制性，给出否定答案并通过记忆强制性符号\(m\)等分析间隙，指出正型记忆非频率均匀强制性，用指数量化缺陷，还表明强制性结构在弱\(*\)收敛下不连续。
AI 中文摘要
我们研究退化情形下具有正型记忆的扩散方程，此时瞬时扩散可能失去强制性。基本问题很简单：完全单调的记忆项能否替代缺失的\(L^{2}(0,\Tend;V)\)强制性？在瞬时能量空间中答案是否定的。通过核的伯恩斯坦表示定义的记忆强制性符号\(m\)来衡量阻碍，当\(k\in L^{1}(0,\infty)\)时\(m\)等于\(\operatorname{Re}\hat{k}\)。对于有限\(L^{1}\)质量的核，一个精确的频率恒等式将瞬时能量与记忆耗散之间的间隙表示为谱权重\(1 - m(\omega)\)；若记忆形式非平凡，当\(\norm{k}_{L^{1}(0,\infty)}\leq1\)时，间隙对所有状态和所有时间视界均非负。在固定视界处，阈值是有限视界强制性轮廓\(\Lambda_{k}(\Tend)\)，其单位穿越定义了一个临界视界，且也适用于无限\(L^{1}\)质量的核，包括分数核。然而，对于每个局部可积的完全单调核，当\(|\omega|\to\infty\)时\(m(\omega)\to0\)。因此，正型记忆是耗散的，但不是频率均匀强制性的：不存在常数\(c>0\)使记忆耗散主导\(c\int_{0}^{\Tend}a_{1}(u,u)\)。这是一个不可行定理，我们通过对每个非常数核有效的强制性间隙指数\(\rho\in[0,2]\)使缺陷量化。最后，在相关时间测度的弱\(*\)收敛下，整个强制性结构是不连续的。由这个不可行结果推动的图空间适定性理论及其目标的认证稳定性在一篇配套论文中展开。
英文摘要
We study Volterra memory terms with locally integrable completely monotone kernels on a finite time interval $(0,\Tend)$ and ask how much $L^{2}$ coercivity they retain at high temporal frequencies. The main tool is an exact formula for the Rayleigh quotients of the cosine modes $ψ_n(t)=(2/\Tend)^{1/2}\cos(2πnt/\Tend)$. It shows that these quotients lie between $(1-κ_n/(2πn))\,m(2πn/\Tend)$ and $m(2πn/\Tend)$, where $m$ is the real memory symbol and $κ_n\in[1-e^{-2πn},1]$ is the best constant valid for all such kernels. Consequently, for non-constant kernels the algebraic decay rate of the quotients does not depend on $\Tend$ and coincides with the decay index $ρ\in[0,2]$ of $m$; any value in $[0,2]$ occurs. The Gaussian kernel $e^{-t^{2}}$, which is of positive type but not completely monotone, shows that this may fail otherwise: its quotients decay like $n^{-4}$, while its symbol decays faster than any power. We also show that the largest Rayleigh quotient on $(0,\Tend)$ is a continuous and strictly increasing function of $\Tend$, so that a kernel of total mass larger than one has exactly one critical horizon. Finally, because the memory operator on $(0,\Tend)$ is compact, it does not contribute to the uniform $L^{2}$ coercivity constant, and for $k_n(t)=ne^{-nt}$ the associated operators converge to the identity strongly but not in norm. The diffusion equation with memory serves as a model throughout.
