This paper is devoted to the analysis of some problems of stabilization of fractional (in time) partial differential equations with memory. The fractional derivative that we consider here is the Caputo derivative which depends on two parameters \(\alpha \in (0,1)\) and \(\eta >0\). Our study is concerned with two different kinds of systems with memory. More precisely, we show that the presence of the memory for the first model improves the behavior of the energy, but in the second model it seems that it is not enough to make the energy decrease; that is why we add a damping term and provide a polynomial stabilization.

中文翻译：

---

具有记忆的分数进化系统的稳定化

本文致力于分析带记忆的分数阶（时间）偏微分方程稳定化的一些问题。我们在这里考虑的分数导数是Caputo导数，它取决于两个参数\（\ alpha \ in（0,1）\）和\（\ eta> 0 \）。我们的研究涉及两种不同类型的内存系统。更确切地说，我们表明第一个模型的内存的存在可以改善能量的行为，但是在第二个模型中，似乎不足以使能量减少；第二个模型似乎不足以减少能量。这就是为什么我们添加阻尼项并提供多项式稳定的原因。