Ricerche di Matematica ( IF 1.1 ) Pub Date : 2021-07-07 , DOI: 10.1007/s11587-021-00617-w Hocine Makheloufi 1, 2 , Mounir Bahlil 1, 2
In this research work, we consider the second-order viscoelastic equation with a weak internal damping, a time-varying delay term and nonlinear weights
$$\begin{aligned} u_{tt}(t) + {\mathcal {A}} u(t) - \int _0^t g (t-s) {\mathcal {A}} u(s) ds + \mu _1 (t) u_t (t)+ \mu _2(t) u_t(t-\tau (t)) =0 \forall t>0 , \end{aligned}$$together with suitable initial conditions. We first prove the existence of a unique global weak solution by means of the classical Faedo–Galerkin method. Then, by assuming the general condition:
$$\begin{aligned} g'(t) \le - \xi (t) H(g(t)), \forall t\ge 0, \end{aligned}$$where H is a positive increasing and convex function and \(\xi \) is a positive function which is not necessarily monotone, we establish optimal explicit and general stability estimates which rely on the well-known multipliers method and some properties of convex functions. This study generalizes and improves many earlier ones in the existing literature.
中文翻译：
具有非常量延迟项和非线性权重的抽象粘弹性方程的全局适定性和稳定性结果
在这项研究工作中，我们考虑具有弱内部阻尼、时变延迟项和非线性权重的二阶粘弹性方程
$$\begin{aligned} u_{tt}(t) + {\mathcal {A}} u(t) - \int _0^tg (ts) {\mathcal {A}} u(s) ds + \mu _1 (t) u_t (t)+ \mu _2(t) u_t(t-\tau (t)) =0 \forall t>0 , \end{aligned}$$加上合适的初始条件。我们首先通过经典的 Faedo-Galerkin 方法证明了唯一全局弱解的存在。然后，通过假设一般条件：
$$\begin{aligned} g'(t) \le - \xi (t) H(g(t)), \forall t\ge 0, \end{aligned}$$其中H是一个正递增的凸函数，\(\xi \)是一个不一定单调的正函数，我们建立了最优的显式和一般稳定性估计，它依赖于众所周知的乘法器方法和凸函数的一些性质。本研究对现有文献中的许多早期研究进行了概括和改进。
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