In this paper, we are concerned with the following problem:
$$\begin{aligned} & u_{tt} + \alpha\Delta^{2} u -\Delta_{p} u - \int_{-\infty}^{t} g(t-s) \Delta^{2} u (s) \,d s - \mu_{1} \Delta u_{t} \\ &\quad{} -\mu_{2} \Delta u_{t} \bigl(t-\tau(t)\bigr) +f(u) = h,\quad \Omega\times {\mathbb {R}}^{+} , \end{aligned}$$
(1.1)
$$\begin{aligned} & u = \Delta u= 0\quad \text{on } \partial\Omega \times {\mathbb {R}}^{+} , \end{aligned}$$
(1.2)
$$\begin{aligned} & u(x,0) = u_{0} (x) , \qquad u_{t} (x,0) = u_{1} (x) ,\quad x\in \Omega, \end{aligned}$$
(1.3)
$$\begin{aligned} & u_{t} (x, t)=f_{0} (x, t),\quad (x, t) \in\Omega\times\bigl[- \tau(0), 0\bigr) , \end{aligned}$$
(1.4)
where Ω is a bounded domain of \({\mathbb {R}}^{n}\) with a sufficiently smooth boundary ∂Ω, and
$$\Delta_{p} u=\operatorname{div}\bigl( \Vert \nabla u \Vert ^{p-2} \nabla u \bigr) $$
is the p-Laplacian operator. The unknown function \(u(x,t)\) denotes the transverse displacement of a plate filament with prescribed history \(u_{0} (x,t), t\leq0 \). The constants α and \(\mu_{1}\) are positive and \(\mu_{2}\) is a real number. The function \(\tau(t)>0\) represents the time-varying delay, \(g>0\) is the memory kernel and f is forcing term.
The plate equation with lower order perturbation of p-Laplacian type,
$$u_{tt} +\Delta^{2} u -\Delta_{p} u -\Delta u_{t} = h(x), $$
has been extensively studied (see [1–3]) and results concerning existence, nonexistence and long-time behavior of solutions have been considered. This model can be regarded as describing elastoplastic microstructure flows. On the other hand, the elliptic problem for p-Laplace operator can be found in [4]. Recently, Torres [5] showed the existence of a solution for the fractional p-Laplacian Dirichlet problem with mixed derivatives.
When \(\mu_{2}=0\) in Eq. (1.1), that is, in the absence of delay, problem (1.1) with strong damping was investigated by Jorge Silva and Ma [6]. They established exponential stability of solutions under the condition
$$\begin{aligned} g'(t)\leq-cg(t), \quad\forall t\geq0, \end{aligned}$$
(1.5)
for some \(c>0\). Andrade et al. [7] proved exponential stability of solutions for the plate equation with finite memory and p-Laplacian. The viscosity term \(-\Delta u_{t}\) is often called a Kelvin–Voigt type dissipation or strong dissipation; it appears in phenomena of wave propagation in a viscoelastic material. Nakao [8] obtained the existence and uniqueness of a global decaying solution for the quasilinear wave equation with Kelvin–Voigt dissipation and a derivative nonlinearity. Pukach et al. [9] studied sufficient conditions of nonexistence of global in time solution for a nonlinear evolution equation with memory generalizing the Voigt–Kelvin model. Recently, Cavalcanti et al. [10] considered intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density.
Time delays so often arise in many physical, chemical, biological, thermal, and economical phenomena because these phenomena depend not only on the present state but also on the past history of the system in a more complicated way. In recent years, there has been published much work concerning the wave equation with constant delay or time-varying delay effects. Nicaise and Pignotti [11] investigated some stability results for the following wave equation with a linear damping and delay term in the domain:
$$\begin{aligned} u_{tt } -\Delta u +\mu_{1} u_{t} + \mu_{2} u_{t}(t-\tau)=0 \end{aligned}$$
(1.6)
in the case \(0<\mu_{2} <\mu_{1} \). Moreover, the same results were obtained when both the damping and the delay act on the boundary. Nicaise and Pignotti [12] studied exponential stability results for the following wave equation with time-dependent delay:
$$\begin{aligned} u_{tt } -\Delta u +\mu_{1} u_{t} + \mu_{2} u_{t}\bigl(t-\tau(t)\bigr)=0 \end{aligned}$$
(1.7)
under the condition \(\vert \mu_{2} \vert < \sqrt{1-d} \mu_{1} \). Kirane and Said-Houari [13] considered the following viscoelastic wave equation with a linear damping and a delay term:
$$\begin{aligned} u_{tt } -\Delta u + \int_{0}^{t} g(t-s) \Delta u(s) \,ds +\mu_{1} u_{t} + \mu_{2} u_{t} (t-\tau) =0, \end{aligned}$$
(1.8)
where \(\mu_{1}\) and \(\mu_{2}\) are positive constants. When \(\mu_{2} \leq\mu_{1}\), they proved general decay of the energy under the condition
$$\begin{aligned} g'(t) \leq-\xi(t) g(t), \quad\forall t\geq0, \end{aligned}$$
(1.9)
where \(\xi:R^{+} \rightarrow R^{+}\) is a nonincreasing differentiable function. Dai and Yang [14] improved the results of [13]. They also obtained an exponentially decay results for the energy of the problem (1.8) in the case \(\mu_{1}=0\). Liu [15] studied a general decay result for the following viscoelastic wave equation with time-dependent delay:
$$\begin{aligned} u_{tt } -\Delta u + \int_{0}^{t} g(t-s) \Delta u(s) \,ds +\mu_{1} u_{t} + \mu_{2} u_{t} \bigl(t-\tau(t)\bigr) =0 \end{aligned}$$
(1.10)
under the conditions (1.9) and \(\vert \mu_{2} \vert < \sqrt{1-d} \mu_{1}\). For the plate equation with time delay term, Yang [16] considered the stability for an Euler–Bernoulli viscoelastic equation with constant delay
$$\begin{aligned} u_{tt } +\Delta^{2} u + \int_{0}^{t} g(t-s) \Delta^{2} u(s) \,ds + \mu_{1} u_{t} + \mu_{2} u_{t} (t-\tau) =0 \end{aligned}$$
(1.11)
under the conditions (1.5) and \(0< \vert \mu_{2} \vert < \mu_{1}\). Moreover, he proved the exponential decay results of the energy in the case \(\mu_{1}=0\). Recently, Feng [17] investigated an exponential stability results for the following plate equation with time-varying delay and past history:
$$\begin{aligned} u_{tt } + \alpha\Delta^{2} u - \int_{-\infty}^{t} g(t-s) \Delta^{2} u(s) \,ds + \mu_{1} u_{t} + \mu_{2} u_{t} \bigl(t- \tau(t)\bigr) +f(u)=0 \end{aligned}$$
(1.12)
under the conditions (1.5) and \(0< \vert \mu_{2} \vert < \sqrt{1-d} \mu _{1}\). Mustafa and Kafini [18] showed the decay rates for memory type plate system (1.12) with \(\tau(t)=\tau\) and \(f(u)=-u \vert u \vert ^{\gamma}\). Park [19] obtained the general decay estimates for a viscoelastic plate equation with time-varying delay under the condition (1.9). The stability of the solutions to a viscoelastic system under the condition (1.9) was studied in [20–23] and the references therein. With respect to wave equation with strong time delay, there is just little published work. Messaoudi et al. [24] considered the following wave equation with strong time delay:
$$\begin{aligned} u_{tt } -\Delta u -\mu_{1} \Delta u_{t} - \mu_{2} \Delta u_{t} (t-\tau) =0 \end{aligned}$$
(1.13)
and proved the well-posedness under the condition \(\vert \mu _{2} \vert \leq \mu_{1}\) and obtained exponential decay of energy under the condition \(\vert \mu_{2} \vert < \mu_{1} \). Recently, Feng [25] established the general decay result for the following viscoelastic wave equation with strong time-dependent delay:
$$\begin{aligned} u_{tt } -\Delta u + \int_{0}^{t} g(t-s) \Delta u(s) \,ds -\mu_{1} \Delta u_{t} - \mu_{2} \Delta u_{t} \bigl(t-\tau(t) \bigr) =0 \end{aligned}$$
(1.14)
under the conditions (1.9) and \(\vert \mu_{2} \vert < \sqrt {1-d} \mu_{1}\).
However, to the best of my knowledge, there is no stability result for the viscoelastic plate equation with strong time-varying delay. Motivated by [24, 25], we study a general decay result for viscoelastic plate equation with p-Laplacian and time-varying delay (1.1)–(1.4) for relaxation function g satisfying the condition (1.9). This result improves on earlier ones in the literature because it allows for certain relaxation functions which are not necessarily of exponential or polynomial decay.
We end this section by establishing the usual history setting of problem (1.1)–(1.4). Following a method devised in [26–29], we shall use a new variable \(\eta^{t}\) to the system with past history. Let us define
$$\begin{aligned} \eta= \eta^{t} (x, s) = u(x, t) -u(x, t-s), \quad (x, s)\in\Omega\times { \mathbb {R}}^{+} , t\geq0 . \end{aligned}$$
(1.15)
Differentiating in (1.15) we have
$$\begin{aligned} \eta_{t}^{t} (x, s) + \eta_{s}^{t} (x, s) = u_{t}(x, t) , \quad (x, s)\in\Omega\times{\mathbb {R}}^{+} , t\geq0 . \end{aligned}$$
Taking \({\alpha= 1+ \int_{0}^{\infty}g (s) \,d s}\), the original problem (1.1)–(1.4) can be transformed into the new system
$$\begin{aligned} \textstyle\begin{cases} u_{tt}+ \Delta^{2} u -\Delta_{p} u + \int_{0}^{\infty} g(s) \Delta^{2} \eta^{t} (s) \,ds\\ \quad{} -\mu_{1}\Delta u_{t} - \mu_{2} \Delta u_{t} ( t-\tau(t)) + f(u) =h ,\quad\text{in } \Omega\times{\mathbb {R}}^{+}, \\ \eta^{t}_{t} = -\eta^{t}_{s} +u_{t} , \quad (x, t,s) \in\Omega\times{\mathbb {R}}^{+} \times{\mathbb {R}}^{+}, \\ u_{t}(x, t)=f_{0} (x, t) ,\quad (x, t) \in\Omega\times[- \tau(0), 0), \end{cases}\displaystyle \end{aligned}$$
(1.16)
with boundary conditions
$$\begin{aligned} u(x, t) =0 \quad\text{on }\partial\Omega\times {\mathbb {R}}^{+},\qquad \eta=0 \quad\text{on }\partial \Omega\times{\mathbb {R}}^{+} \times{\mathbb {R}}^{+}, \end{aligned}$$
(1.17)
and initial conditions
$$\begin{aligned} u(x,0) = u_{0} (x),\qquad u_{t} (x,0) = u_{1}(x),\qquad \eta^{t} (x, 0)=0, \qquad\eta^{0} (x, s) =\eta_{0} (x, s) , \end{aligned}$$
(1.18)
where
$$\begin{aligned} \textstyle\begin{cases} u_{0} (x) = u_{0} (x, 0),\quad x\in\Omega, \\ u_{1} (x) = \partial_{t} u_{0} (x, t)|_{t=0},\quad x\in\Omega, \\ \eta_{0} (x, s) = u_{0} (x, 0 ) - u_{0} ( x, -s) , \quad(x, s) \in\Omega \times{\mathbb {R}}^{+}. \end{cases}\displaystyle \end{aligned}$$
The paper is organized as follows. In Sect. 2, we state the notation and main result. In Sect. 3, we prove the general decay of the solutions to the viscoelastic plate equation with p-Laplacian and time-varying delay by using the energy perturbation method.