General decay for viscoelastic plate equation with p-Laplacian and time-varying delay

In this paper, we are concerned with the following problem: where Ω is a bounded domain of \({\mathbb {R}}^{n}\) with a sufficiently smooth boundary ∂Ω, and 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, 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 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: 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: 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: 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 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: 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 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: 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: 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: under the conditions (1.9) and \(\vert \mu_{2} \vert < \sqrt {1-d} \mu_{1}\).

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 Differentiating in (1.15) we have Taking \({\alpha= 1+ \int_{0}^{\infty}g (s) \,d s}\), the original problem (1.1)–(1.4) can be transformed into the new system with boundary conditions and initial conditions where 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.

For a Banach space \(X, \Vert \cdot \Vert _{X}\) denotes the norm of X. For simplicity, we denote \(\Vert \cdot \Vert _{L^{2}(\Omega)}\) by \(\Vert \cdot \Vert \). We use the standard Lebesgue and Sobolev spaces, with their usual scalar products and norms, and the Sobolev–Poincaré inequality for \(q\geq2\) if \(1\leq n\leq4\) or \(2\leq q \leq\frac{2n}{n-4}\) if \(n\geq5\).

In the following, we fix some notations on the function spaces that will be used. Let and In order to consider the new variable η, we introduce the weighted \(L^{2}\)-spaces which are Hilbert spaces endowed with the inner products and norms respectively. To simplify the notation, we define the Hilbert spaces

Let us begin with the precise hypotheses on the constant p and the functions f and g. For \(n\in{\mathbb {N}}\), we assume that Then The nonlinear function \(f: {\mathbb {R}} \rightarrow{\mathbb {R}}\) satisfying \(f(0)=0\) and the growth condition, where \(k_{0} >0\) and which implies that \(H^{2} (\Omega) \hookrightarrow L^{2(\rho+1)} (\Omega)\). In addition, we assume that where \(F(z)=\int_{0}^{z} f(s) \,d s \).

For the relaxation function g, we assume that \(g:{\mathbb {R}}^{+} \rightarrow{\mathbb {R}}^{+}\) is a nonincreasing \(C^{1}\) function satisfying and there exists a nonincreasing differentiable function \(\xi: {\mathbb {R}}^{+} \rightarrow{\mathbb {R}}^{+}\) satisfying and As in [12], for the time-varying delay, we assume that \(\tau\in W^{2, \infty}([0, T])\) for \(T>0\), and there exist positive constants \(\tau_{0}, \tau_{1}\) and d satisfying and that \(\mu_{1}\) and \(\mu_{2}\) satisfy

In order to state our main result, we define the energy of problem (1.16)–(1.18) by where ξ and λ are positive constants satisfying Note that this choice of ξ is possible from assumption (2.9).

Now, let us define the perturbed modified energy by where Then it is easily shown that there exists \(C_{1}>0\) such that where \(C_{1} = \max\{ 1, C_{*} \}\).