General decay for a viscoelastic wave equation with strong time-dependent delay

In this paper, we consider the following viscoelastic wave equation with a strong damping and a strong time-dependent delay in the internal feedback: where \(x\in\Omega\), and \(\Omega\subseteq {\mathbb{R}}^{n}\) (\(n\geq1\)) is a bounded domain with smooth boundary ∂Ω. The function \(g(t)\) is the relaxation function. \(\mu_{1}\), \(\mu_{2}\) are constants and \(\tau(t)>0\) denotes the time-dependent delay.

We consider the following initial conditions: and the following boundary conditions:

Recently the control of wave equations with time delay effects has become an active area of research. The delay effects often appear in many practical problems and may turn a well-behaved system into a wild one. The presence of delay can be a source of instability. Here we mention the work of Nicaise and Pignotti [1]. In this work the authors considered a wave equation with time delay of the form Under the assumption \(0<\mu_{2}<\mu_{1}\), they proved exponential stability of the system. For viscoelastic wave equation with time delay, Kirane and Said-Houari [2] studied the following equation: and obtained the global well-posedness of solutions and established the energy decay under the assumption \(0<\mu_{2}\leq\mu_{1}\). Liu [3] considered a wave with time-dependent delay and established the general decay of the system under \(|\mu_{1}|<\sqrt{1-d}\mu_{1}\). Dai and Yang [4] considered the same equation as in [2] and improved the above results under weaker conditions. In this work, the authors obtained the global well-posedness without any restrictions on \(\mu_{1}\), \(\mu_{2}\), and established an exponential decay result of energy in the case \(\mu_{1}=0\). Benaissa et al. [5] investigated a wave equation with nonlinear time delay of the form They proved the global existence of solution under assumption of a relation between the weight of the delay term in the feedback and the weight of the term without delay. In addition, they obtained the general decay of energy. Liu and Zhang [6] considered a wave equation with past history and time delay in internal feedback They proved the global well-posedness without any restrictions on \(\mu_{1}\), \(\mu_{2}\). Furthermore, they also proved the exponential decay of energy with \(0<|\mu_{2}|<\mu_{1}\). Kafini et al. [7] consider a nonlinear damped second-order evolution equation with delay and they proved that the energy of the solutions blows up in finite time under some suitable assumptions. Recently, Alabau-Boussouira et al. [8] studied a wave equation with past history and time delay They showed that the system is exponentially stable if the coefficient of delay k is small enough. They also established the stability in the case \(\tau=0\) and \(k<0\). For some more results concerning the wave equation with a weak time delay term under an appropriate assumption between \(\mu_{1}\) and \(\mu_{2}\), one can refer to Benaissa et al. [9], Datko et al. [10], Liu [11], Nicaise and Pignotti [12–14], Nicaise et al. [15], Nicaise and Valein [16], Xu et al. [17], and the references therein. With respect to waves with strong time delay there is just little published work. The only one we found is duo to Messaoudi et al. [18]. In this work, they investigated the following equation: and proved the well-posedness under the assumption \(|\mu_{2}|\leq\mu_{1}\) and established exponential decay of energy under the assumption \(|\mu_{2}|<\mu_{1}\). Moreover, they also studied a wave equation with distributed delay.

For the relaxation function g, we assume \(g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) is a nonincreasing \(C^{1}\) function satisfying and there exists a nonincreasing differentiable function \(\zeta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) satisfying, for \(t\geq0\), and

Concerning the delay \(\tau(t)\), we assume where the constants \(\tau_{0}\) and \(\tau_{1}\) are two positive constants. We assume further that

The weak solutions of (1.1)-(1.3) are defined as follows: for given initial data \((u_{0},u_{1})\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)\), we call a function \(U=(u,u_{t})\in C(\mathbb{R}^{+},H_{0}^{1}(\Omega)\times L^{2}(\Omega))\) a weak solution to the problem (1.1)-(1.3) if \(U(0)=(u_{0},u_{1})\) and for all \(\omega\in H_{0}^{1}(\Omega)\).

The energy functional of problem (1.1)-(1.3) is defined by where \(\xi>0\) is a constant to be determined later, the constant \(\lambda>0\), as below, has been introduced in [13], and we have

Now we define the Lyapunov functional \(F(t)\) by where \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are positive constants to be taken later. First we know that, for \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) small enough, there exist two positive constants \(\beta_{1}\) and \(\beta_{2}\) such that, for any \(t>0\),