Our main goal in the present work is to address an integro-differential model under localized viscoelastic and frictional effects arising in the Boltzmann theory of viscoelasticity. More precisely, we consider a general version in the history context of the pioneer localized viscoelastic problem approached by Cavalcanti and Oquendo (2003) in the null history scenario, and more recently by Cavalcanti et al. (2018) in the history framework. By means of a new observability inequality, we prove a general stability result to the model under a weaker assumption on the localized frictional damping and a slower condition on the decreasing memory kernel (of polynomial type) than the previously mentioned works. To achieve such stability results, we still work in a general setting by removing the assumption on complementary damping mechanisms and show, in some reasonable situations concerning the density coefficient, that the localized viscoelastic effect is enough to ensure the general stability (of polynomial type) to the problem.

我们当前工作的主要目标是解决在玻尔兹曼粘弹性理论中产生的局部粘弹性和摩擦作用下的积分微分模型。更精确地讲，我们在历史背景下考虑了Cavalcanti和Oquendo（2003）在零历史情况下以及更近期的Cavalcanti等人提出的先驱性局部粘弹性问题的一般版本。（2018）在历史框架中。通过一个新的可观察性不等式，我们证明了该模型的一般稳定性结果是在与先前提到的工作相比，在较弱假设的局部摩擦阻尼和较慢的条件（递减的内存核（多项式类型））下进行的。为了获得这样的稳定​​性结果，我们仍然在一般情况下通过删除以下假设进行工作：补充阻尼机制，并在某些合理的情况下（与密度系数有关）表明，局部粘弹性效应足以确保问题的总体稳定性（多项式类型）。