This work considers the Galerkin approximation and analysis for a hyperbolic integrodifferential equation, where the non-positive variable-sign kernel and nonlinear-nonlocal damping with both the weak and viscous damping effects are involved. We derive the long-time stability of the solution and its finite-time uniqueness. For the semi-discrete-in-space Galerkin scheme, we derive the long-time stability of the semi-discrete numerical solution and its finite-time error estimate by technical splitting of intricate terms. Then we further apply the centering difference method and the interpolating quadrature to construct a fully discrete Galerkin scheme and prove the long-time stability of the numerical solution and its finite-time error estimate by designing a new semi-norm. Numerical experiments are performed to verify the theoretical findings.

翻译：本文研究一类双曲型积分微分方程的Galerkin逼近及其分析，该方程涉及非正变号核以及兼具弱阻尼与黏性阻尼效应的非线性非局部阻尼项。我们推导了解的长时间稳定性及其有限时间唯一性。针对空间半离散Galerkin格式，通过对复杂项进行技术性分解，得到了半离散数值解的长时间稳定性及其有限时间误差估计。进一步，我们应用中心差分法与插值型求积公式构建了全离散Galerkin格式，并通过设计新的半范数证明了数值解的长时间稳定性及其有限时间误差估计。数值实验验证了理论结果。