In this paper, we consider an abstract second-order viscoelastic equation with infinite memory, time-varying delay and a nonlinear source term. To the best of our knowledge, there is no blow-up result of solutions for the nonlinear viscoelastic equation with time-varying delay. Moreover, our result extends the blow-up result obtained for problems without memory term to those with infinite memory and problems with constant delay to time-varying delay. Under suitable conditions on initial data, the kernel memory function and the involved functionals, we establish the local existence using the semigroup theory and we prove a finite time blow-up result for the solution with positive initial energy. We also give some applications to illustrate our main result.

在本文中，我们考虑了一个具有无限记忆、时变延迟和非线性源项的抽象二阶粘弹性方程。据我们所知，具有时变延迟的非线性粘弹性方程的解没有爆炸结果。此外，我们的结果将没有记忆项的问题扩展到具有无限记忆的问题，将具有恒定延迟的问题扩展到时变延迟。在初始数据、核记忆函数和相关泛函的适当条件下，我们利用半群理论建立了局部存在性，并证明了具有正初始能量的解的有限时间爆破结果。我们还给出了一些应用程序来说明我们的主要结果。