This paper develops a unified method to derive compactness properties and convergence to a steady state as well as decay estimates of global bounded solutions of the abstract semilinear second order integro-differential equation

with finite (\(\tau =t\)) or infinite (\(\tau =\infty \)) memory. Here, A is a continuous linear operator, k is a positive nonincreasing function, f is a nonlinear function such that \(A+f\) is the gradient of a potential satisfying the Łojasiewicz–Simon inequality, and the time-dependent right-hand side g decays to 0 at infinity. The general results are applied to particular semilinear evolution viscoelastic problems at the end of the paper.

本文开发了一种统一的方法来推导紧凑性和收敛到稳态以及抽象半线性二阶积分微分方程的全局有界解的衰减估计

具有有限（\(\tau =t\)）或无限（\(\tau =\infty \)）内存。这里，A是一个连续线性算子，k是一个正的非增函数，f是一个非线性函数，使得\(A+f\)是满足 Łojasiewicz–Simon 不等式的势的梯度，以及与时间相关的右-手边g在无穷远处衰减到 0。论文最后将一般结果应用于特定的半线性演化粘弹性问题。