The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely, \(g^{\prime }(t)\le -\xi (t) G(g(t))\). We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = s^p and p covers the full admissible range [1,2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature.

这项工作的主要目的是研究粘弹性方程的长期行为，该方程具有对数源项和局部边界上的非线性反馈。在潜力井的框架中，我们首先显示全球存在。然后，我们在松弛函数g的行为的一个非常笼统的假设下讨论该问题的渐近行为，即\（g ^ {\ prime}（t）\ le-\ xi（t）G（g（t ））\）。我们建立了明确的和一般的衰减结果，当G（s）= s ^p和p时，我们可以从中恢复众所周知的指数和多项式速率涵盖了整个允许范围[1,2）。我们的结果是在没有对边界阻尼项施加任何限制性增长假设的情况下获得的。这项工作概括并改进了文献中的许多早期结果。