Well-posedness and asymptotic behaviour of a wave equation with non-monotone memory kernel

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.