In this paper, we consider a nonlinear viscoelastic equation with nonlocal nonlinear boundary damping and two nonlinear source terms (boundary and interior). We obtain the global existence of solution via the potential well method. By introducing suitable Lyapunov functionals, using the multiplier method, and constructing convex differential inequality, we establish an explicit and general decay rate result. This improves previous decay results concerning the problem. We also get a finite time blow-up result of solution with positive initial energy under suitable conditions on the initial data and positive memory function. This is the first result for blow-up of the problem with nonlocal nonlinear boundary damping.

中文翻译：

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具有非局部非线性边界耗散的粘弹性方程解的能量衰减和爆破

在本文中，我们考虑具有非局部非线性边界阻尼和两个非线性源项（边界和内部）的非线性粘弹性方程。我们通过势阱法获得解的全局存在性。通过引入合适的李雅普诺夫泛函，使用乘法器方法，并构造凸微分不等式，我们建立了显式和一般的衰减率结果。这改进了先前关于该问题的衰减结果。在初始数据和正记忆函数的适当条件下，我们还得到了具有正初始能量的解的有限时间膨胀结果。这是解决非局部非线性边界阻尼问题的第一个结果。