High-frequency coercivity loss for completely monotone memory on bounded time intervals

Abstract

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 ${\kappa }_n\in [1-e^{-2\pi n},1]$ 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 $\rho \in [0,2]$ 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 $k_n(t)=n{e}^{-nt}$ the associated operators converge to the identity strongly but not in norm. The diffusion equation with memory serves as a model throughout.