Given a positive operator A on some Hilbert space, and a nonnegative decreasing summable function \(\mu \), we consider the abstract equation with memory

$$\begin{aligned} \ddot{u}(t)+ A u(t)- \int _0^t \mu (s)Au(t-s) ds=0 \end{aligned}$$

modeling the dynamics of linearly viscoelastic solids. The purpose of this work is to provide numerical evidence of the fact that the energy

$$\begin{aligned} {{\textsf{E}}}(t)=\Big (1-\int _0^t\mu (s)ds\Big )\Vert u(t)\Vert ^2_1+\Vert \dot{u}(t)\Vert ^2 +\int _0^t\mu (s)\Vert u(t)-u(t-s)\Vert ^2_1ds \end{aligned}$$

of any nontrivial solution cannot decay faster than exponential, no matter how fast might be the decay of the memory kernel \(\mu \). This will be accomplished by simulating the integro-differential equation for different choices of the memory kernel \(\mu \) and of the initial data.

中文翻译：

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线性粘弹性中缺乏超稳定轨迹：一种数值方法

给定某个希尔伯特空间上的正算子A和一个非负递减可求和函数\(\mu \)，我们考虑具有记忆的抽象方程

$$\begin{aligned} \ddot{u}(t)+ A u(t)- \int _0^t \mu (s)Au(ts) ds=0 \end{aligned}$$

模拟线性粘弹性固体的动力学。这项工作的目的是提供能量的事实的数字证据

$$\begin{aligned} {{\textsf{E}}}(t)=\Big (1-\int _0^t\mu (s)ds\Big )\Vert u(t)\Vert ^2_1+\垂直 \dot{u}(t)\Vert ^2 +\int _0^t\mu (s)\Vert u(t)-u(ts)\Vert ^2_1ds \end{aligned}$$

任何非平凡解的衰减速度都不能快于指数，无论内存内核\(\mu \)的衰减速度有多快。这将通过针对内存内核\(\mu \)和初始数据的不同选择模拟积分微分方程来实现。