In this paper, we consider the nonlinear abstract equation

$$\begin{aligned} u_{tt}+Au-\int _{0}^{t}g(t-s)Au(s)\mathrm{d}s+h(u_{t})=j(u) \end{aligned}$$

subject to a competing effect of viscoelastic and frictional dampings. With very general assumptions on the behavior of g at infinity and the behavior of h near 0, we establish explicit and optimal energy decay result. To the best of our knowledge, this is the first time we have such combination of generality and optimality in one explicit formula for the energy decay rates of this system.

中文翻译：

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非线性抽象粘弹性耗散系统的最佳能量衰减结果

在本文中，我们考虑了非线性抽象方程

$$ \ begin {aligned} u_ {tt} + Au- \ int _ {0} ^ {t} g（ts）Au（s）\ mathrm {d} s + h（u_ {t}）= j（u ）\ end {aligned} $$

受到粘弹性和摩擦阻尼的竞争影响。通过对g在无穷大处的行为和h在0附近的行为进行非常一般的假设，我们建立了明确且最佳的能量衰减结果。据我们所知，这是我们第一次在一种明确的公式中将通用性和最优性结合起来，用于该系统的能量衰减率。