This paper concerns a nonlinear viscoelastic wave equation with time-dependent delay. Under suitable relation between the weight of the delay and the weight of the term without delay, we prove the global existence of weak solutions by the combination of the Galerkin method and potential well theory. In addition, by assuming the minimal conditions on the $L^1(0,\infty)$ relaxation function $g$, namely, $g'(t)\leq-\xi(t)H(g(t))$, where $H$ is an increasing and convex function and $\xi$ is a nonincreasing differentiable function, and by using some properties of convex functions, we establish optimal explicit and general energy decay results. This result is new and substantially improves existing results in the literature.

中文翻译：

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非线性时滞粘弹性波动方程的最优衰减率

本文涉及具有时变时滞的非线性粘弹性波动方程。在延迟权重与无延迟项权重之间的适当关系下，我们通过结合Galerkin方法和势阱理论证明了弱解的整体存在性。另外，通过假设$ L ^ 1（0，\ infty）$松弛函数$ g $的极小条件，即$ g'（t）\ leq- \ xi（t）H（g（t）） $，其中$ H $是一个递增和凸函数，而$ \ xi $是一个非递增微分函数，并且通过使用凸函数的某些属性，我们建立了最佳的显式能量衰减和一般能量衰减结果。该结果是新的，并且大大改善了文献中的现有结果。