In this paper, we study the well-posedness and stability of a wave equation with infinitely structural memory, herein the memory kernel function does not satisfy the monotonicity. For the model, the history function space setting is a main difficulty because the usual space setting will lead the shift semigroup to be a unbounded semigroup. In the present paper, we modify the history function space setting and prove the well-posedness of the system. Further we study the stability of the system via Lyapunov function method. By constructing appropriate Lyapunov function, we show that the energy function of the system decays exponentially if the memory kernel function satisfies some conditions. Finally, we give an example of the memory kernel function that is not monotone but satisfies all conditions proposed in the present paper.

在本文中，我们研究了具有无限结构记忆的波动方程的适定性和稳定性，其中记忆核函数不满足单调性。对于模型，历史函数空间设置是一个主要的困难，因为通常的空间设置将导致移位半群成为无界半群。在本文中，我们修改了历史函数的空间设置，并证明了系统的适定性。进一步，我们通过Lyapunov函数方法研究了系统的稳定性。通过构造适当的Lyapunov函数，我们表明，如果内存内核函数满足某些条件，则系统的能量函数将呈指数衰减。最后，我们给出一个不是单调但满足本文提出的所有条件的内存核函数示例。