In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping and a nonlinear source term. Under some suitable assumptions on the weight of the delayed feedback, the weight of the non-delayed feedback and the behavior ofthe relaxation function, we establish two explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some properties of the convex functions. Moreover, we give some applications and examples. This work generalizes the previous results without time delay term to those with delay in the nonlinear damping.

中文翻译：

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希尔伯特空间中具有非线性内阻尼时滞项的抽象非线性粘弹性方程的最优衰减。

在本文中，我们考虑了希尔伯特空间中的二阶抽象粘弹性方程，该方程具有非线性内部阻尼和非线性源项的延迟项。在关于延迟反馈的权重，非延迟反馈的权重和松弛函数的行为的一些适当假设下，我们通过引入适当的Lyaponov泛函和该函数的一些性质来建立能量的两个显式和一般的衰减率结果凸函数。此外，我们给出了一些应用程序和示例。这项工作将没有时间延迟项的先前结果推广到了非线性阻尼中有延迟的那些结果。