GENERAL DECAY OF SOLUTIONS OF A WEAKLY COUPLED ABSTRACT EVOLUTION EQUATIONS WITH ONE FINITE MEMORY CONTROL

In this work, we consider the following abstract evolution system:

$ \begin{equation*} \left\{ \begin{array}{ll} u_{tt}(t)+Au(t)- \int_{0}^{t}g(t-s)A^{\theta }u(s)ds+\alpha v(t)=0, & t>0, \\ v_{tt}(t)+Av(t)+\alpha u(t)=0, & t>0, \\ u(0)=u_{0},\text{ }u_{t}(0)=u_{1},\text{ }v(0)=v_{0},\text{ }v_{t}(0)=v_{1},& \end{array} \right. \end{equation*} $

where $ A:\mathcal{D}(A)\subset H\longrightarrow H $ is a linear positive definite self-adjoint operator, $ H $ is a Hilbert space, $ g $ is a positive non-increasing function with some general decay rate, $ \theta\in \lbrack 0,1] $, $ \alpha $ is a positive constant and $ u_0 ,u_1 ,v_0 $ and $ v_1 $ are fixed initial data. Under appropriate conditions on $ g,\,\alpha $ and the regularity of the initial data, we establish a general decay rate of the solution energy which generalizes some earlier results in the literature. We, also, illustrated our results by performing several numerical tests.