Abstract
In this paper, we consider a weak viscoelastic equation with internal time-varying delay
in a bounded domain. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general decay result for the energy. This work generalizes and improves earlier results in the literature.
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1 Introduction
In this paper, we investigate the following weak viscoelastic equation with a time-varying delay term in the feedback
where \(\varOmega\) is a bounded domain of \(\mathbb{R}^{n}\) (\(n\geq2\)) with a sufficiently smooth boundary \(\partial\varOmega\), \(\alpha\) and \(g\) are positive non-increasing functions defined on \(\mathbb{R}^{+}\), \(\tau (t)>0\) represents the time-varying delay, \(u_{0}, u_{1}, f_{0}\) are given functions belongs to some suitable spaces.
When \(\mu=0\) in the first equation of (1.1), that is in the absence of the delay, problem (1.1) was studied by many authors during the past decades. This type of problem arises in viscoelasticity, especially for the case \(\alpha(t)=1\). We start by mentioning the pioneer works of Dafermos [5, 6], where he discussed a one-dimensional viscoelastic problem, established several existence and asymptotic stability results. For other related works, we refer the readers to [1, 3, 4, 13, 14, 23] and references therein. On the other hand, Messaoudi [17] considered the viscoelastic equation without the delay of the form
under suitable conditions on \(\alpha\) and \(g\). He obtained general stability by making use of the perturbed energy method.
Introducing the delay term makes the problem different from those considered in the literatures. Time delay arises in many applications depending not only on the present state but also on some past occurrences. The presence of delay may be a source of instability. For example, when \(g=0\), it was shown in [8–10, 18, 19, 27] that an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay unless additional control terms have been used. Kirane and Said-Houari [11] considered the problem (1.1) with \(\alpha(t)\equiv1\), an additional linear damping term, \(\mu>0\) and \(\tau(t)\) be a constant delay, and established general decay results under some condition.
The case of the time-varying delay in wave equation has also been studied by several authors, see for example [15, 16, 20–22] and the references therein. See also [12, 26] for the case of the transmission problem with delay.
In the works mentioned above, the authors must used the damping term \(\mu_{1}u_{t}(x, t)\) to control the delay term in the priori estimate of the solution and the decay estimate of the energy. By the way, in [7, 28], the authors improve earlier results in the literature by making using of the viscoelastic term to control the time constant delay term. Similar results have also been obtained for the problem with infinite memory by Guesmia [10]. Alabau-Boussouira et al. [2] considered the following wave delay equation with past history
and established the exponential stability if the coefficient \(k\) is sufficiently small by using a perturbation approach for delay problems first introduced in [24]; the result also holds for the anti-damping, i.e. \(\tau=0\) and \(k<0\). In this aspect, it is worth mentioning the work of Pignotti [25]. In this work, the author considered the problem
and obtained that asymptotic stability is guaranteed if the delay feedback coefficient belongs to \(L^{1}(0, +\infty)\) and the time intervals where the delay feedback is off are sufficiently large.
But, to the best of our knowledge, there is no research on the weak viscoelastic equation (a coefficient \(\alpha(t)\) multiplying the memory term) with time-varying delay. Motivated by there results, we investigate the problem (1.1) under suitable assumptions. Our main contribution is an extension of the previous results from [7, 11, 15, 28] to weak viscoelastic equation without the linear damping term. The plan of this paper is as follows. In Sect. 2, we present some notations and assumptions needed for our work. In Sect. 3, we derive a general decay estimate of the energy.
2 Preliminaries and Main Result
We first introduce some notations that will be used in the proof of our results. We use the standard Lebesgue space \(L^{2}(\varOmega)\) and the Sobolev space \(H^{1}_{0}(\varOmega)\) with their usual scalar products and norms. Throughout this paper, \(C\) and \(C_{i}\) are used to denote the generic positive constant. From now on, we shall omit \(x\) and \(t\) in all functions of \(x\) and \(t\) if there is no ambiguity.
For the relaxation function \(g\) and the potential \(\alpha\), we assumption the following (see [17, 22]):
-
(G1)
\(g, \alpha:\mathbb{R^{+}}\rightarrow R^{+}\) are nonincreasing differentiable functions satisfying
$$ g(0)>0, \quad \int_{0}^{+\infty} g(s)ds< \infty,\qquad \alpha(t)>0,\quad 1-\alpha (t) \int_{0}^{t}g(s)ds\geq l>0. $$(2.1)In addition, we assume that there exists a positive constant \(\alpha_{0}\) such that \(\alpha(t)\geq\alpha_{0}\).
-
(G2)
There exists a nonincreasing differentiable function \(\zeta (t): \mathbb{R^{+}}\rightarrow R^{+}\) satisfying
$$ \zeta(t)>0,\quad g'(t)\leq-\zeta(t)g(t) \qquad \mbox{for}\ t\geq0, \quad \lim_{t\rightarrow+\infty}\frac{-\alpha'(t)}{\zeta(t)\alpha(t)}=0. $$(2.2)
For the time-varying delay \(\tau\), we assume as in [15, 21, 22] that \(\tau\in W^{2, \infty}([0, T])\) for any \(T>0\), and there exist positive constants \(\tau_{0}, \tau_{1}\) and \(d\) such that
The following lemma is concerned with the global well-posedness of the problem (1.1). By using the classical Faedo-Galerkin method, see, e.g. [15, 17, 28], we can prove the Lemma, and we omit the proof here.
Lemma 2.1
Let (2.1)–(2.3) be satisfied. If the initial data \(u_{0}\in H_{0}^{1}(\varOmega), u_{1}\in L^{2}(\varOmega), f_{0}\in L^{2}(\varOmega\times (0, 1))\) and any \(T>0\), then the problem (1.1) has a unique weak solution \((u, u_{t})\in C(0, T; H_{0}^{1}(\varOmega)\times L^{2}(\varOmega) )\) such that
Now, inspired by [15, 17, 21], we define the modified energy functional to the problem (1.1) by
where \(\xi, \lambda\) are suitable positive constants to be determined later, and
First, we fix \(\lambda\) such that
with \(k_{1}\) and \(k_{2}\) being defined in (3.16).
In order give our main theorem, we give the restriction condition on \(\zeta(t)\)
where \(b\) is a positive constant to be chosen in (3.23).
Our main result reads as follows:
Theorem 2.1
Let (G1), (G2) and (2.5) hold. If the coefficient of the time-varying delay satisfies \(|\mu|< a\), then there exist positive constants \(K\) and \(\kappa\) such that the energy of the problem (1.1) satisfies
where \(a\) is positive constant defined by (3.21), which is only dependent on \(g_{0}, l, d\).
3 Energy Decay
In this section, we will prove the energy decay result Theorem 2.1 by constructing an appropriate Lyapunov function. First, we have the following lemmas.
Lemma 3.1
Let (2.1)–(2.3) be satisfied. Then for all regular solution of problem (1.1), the energy function defined by (2.4) satisfies
Proof
Differentiating (2.4) and using the first equation of (1.1) and then integrating by parts, the assumptions (2.1)–(2.3) and some manipulations as in [16, 21], we obtain
Noticing (2.5) and the assumption (G1), (3.1) is established. □
Remark 3.1
Since \((\frac{|\mu|}{2\sqrt{1-d}}+\frac{\xi}{2} )\|u_{t}\|_{2}^{2}\geq0\), \(-\frac{1}{2}\alpha'(t)(\int_{0}^{t}g(s)ds)\|\nabla u\| _{2}^{2}\geq0\), \(E'(t)\) may not be non-increasing.
Now, we define the Lyapunov function
where \(\varepsilon_{i}\), \(i=1, 2\) are two positive real numbers which will be chosen later, and
We can prove that, for sufficiently small \(\varepsilon_{1}, \varepsilon _{2}\), for any \(t\geq0\), the exist two positive constant \(\beta_{1}, \beta_{2}\) such that
The following estimates hold true.
Lemma 3.2
Under the assumption (G1), there exist two positive constants \(C_{1}\) and \(C_{2}\) satisfying
Proof
Differentiating and integrating by parts
Now, Young’s and Poincaré’s inequalities yields (see [16])
and
Choosing \(\delta>0\) sufficiently small and combining the above estimates, we obtain (3.6). □
Lemma 3.3
Under the assumption (G1), we have the following estimate
where \(C_{p}\) is the Poincaré constant.
Proof
The proof of this lemma is similar as Lemma 3.4 in [16]. But we do not have the damping term \(u_{t}(t)\) in this paper. We give the sketch of it. Combining (1.1) and (3.4), we obtain
The first to the third terms on the right hand-side of (3.8) can be estimate as in [17], for any \(\delta>0\), that
Noticing the estimate
where \(C_{p}\) is the Poincaré constant, we obtain
Combining the above estimates with (3.8), we get (3.7). □
Now, we are in the position to prove the general decay result.
Proof of Theorem 2.1
Combining (3.1), (3.6), (3.8) and (G1), after a series of computations, for any \(t\geq t_{0}\), we obtain
By using (3.3), (3.4). Young’s and Poincaré’s inequalities, we obtain
Noticing \(g\) is positive and non-increasing, we have, for any fixed \(t_{0}>0\), for any \(t\geq t_{0}\), that
Hence, (3.9) takes the form
Now, we should deduce the following system of the inequalities
is solvable only if we add some suitable conditions to \(\mu\).
Indeed, we can find solutions of (3.12) according to the following steps.
Step 1. We first take \(\delta\) sufficiently small such that
and
Step 2. Once \(\delta\) is fixed, we select \(\varepsilon_{2}\) sufficiently small such that
Step 3. Then we choose \(\varepsilon_{1}\) satisfying the relation
The choice of \(\varepsilon_{1}\) is possible by the choice of \(\delta\) in (3.13) and (3.14).
From (3.13) and (3.15), we deduce that
Now, let \(k_{1}=\varepsilon_{2}(g_{0}-\delta)-\varepsilon_{1}\) and \(k_{2}=\varepsilon_{1}C_{1}\alpha(0)+\varepsilon_{2}\delta\alpha(0)\). Thus, (3.16) implies that \(k_{1}, k_{2}\) are two positive constants depending on \(g_{0}, l\).
Step 4. Now, we must ensure that the first to the third inequality in (3.12) hold. That is to say, the following system of inequalities
must be solvable.
In fact, by (3.15), the inequality
is obtained. Since \(\lim_{t\rightarrow\infty}\frac{\alpha '(t)}{\alpha(t)}=0\) (which can be deduced from (G2)), we can choose \(t_{1}\geq t_{0}\) so that the third term in (3.17) is obtained for all \(t\geq t_{1}\), and the first term in (3.17) becomes
Step 5. Moreover, by the choice of \(\delta\) in (3.14), we can obtain
Since \(\lambda\) satisfies (2.5), there exists a positive constant \(\xi\) such that
Step 6. To ensure the solvability of the system (3.17), we just need to add a condition given by
here \(a\) is only dependent on \(g_{0}, l, d\). Moreover, the condition (3.21) is possible from (3.20).
Hence, the system (3.12) is solvable. Consequently, there exist two positive constants \(C_{3}\) and \(C_{4}\) such that
Multiplying (3.22) by \(\zeta(t)\) and using (G2), (3.1) and (3.18), for \(t\geq t_{1}\), we obtain
Since \(\zeta(t)\) is nonincreasing, we have
Observing from the definition of \(E(t)\) and assumption (2.1) that
we get
Since \(\lim_{t\rightarrow\infty}\frac{-\alpha'(t)}{\zeta(t)\alpha (t)}=0\), we can choose \(t_{*}\geq t_{1}\) such that
Hence, if we let
that is \(\zeta(t)>b=\frac{2k_{1}}{C_{5}}\geq\frac{2k_{1}\alpha_{0}}{C_{5}\alpha(t)}\), we arrive at
Finally, let \(\mathcal{L}(t)=\zeta(t)L(t)+2C_{4}(t)E(t)\), then we can see that \(\mathcal{L}(t)\) is equivalent to \(E(t)\). Hence, we arrive at
Integrating this over \((t_{*}, t)\), we can deduce that
Consequently, the equivalent relations of \(E(t)\), \(L(t)\) and \(\mathcal{L}(t)\) give the desired result (2.7). □
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Acknowledgements
The authors would like to thank the referees for the careful reading of this paper and for the valuable suggestions to improve the presentation and the style of the paper. This project is supported by Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (Grant No. 15A110017) and the Basic Research Foundation of Henan University of Technology, China (No. 2013JCYJ11).
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Liu, G., Diao, L. Energy Decay of the Solution for a Weak Viscoelastic Equation with a Time-Varying Delay. Acta Appl Math 155, 9–19 (2018). https://doi.org/10.1007/s10440-017-0142-1
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DOI: https://doi.org/10.1007/s10440-017-0142-1