Abstract
This paper deals with the global existence of solutions in a bounded domain for nonlinear viscoelastic Kirchhoff system with a time varying delay by using the energy and Faedo–Galerkin method with respect to the delay term weight condition in the feedback and the delay speed. Furthermore, by using some convex functions properties, we prove a uniform stability estimate.
1. Introduction
1.1. Model
Consider the following viscoelastic Kirchhoff system:where is a bounded domain in , nϵ IN∗, with a smooth boundary , , , and are positive real numbers, and are positive functions which decay exponentially, is a time varying delay, and the initial data are in a suitable function space. is a -function for , with , and .
Time delay is often present in applications and practical problems. In last few years, the control of PDEs with time delay effects has become an active area of research (see, for example, [1–4] and the references therein). In [5], the authors showed that a small delay in a boundary control could turn a well-behaved hyperbolic system into a wild one and therefore delay becomes a source of instability. However, sometimes it can also improve the performance of the system.
By using the Faedo–Galerkin method, Wu in [6] proved the result of local existence and established the decay result by suitable Lyapunov functionals according to appropriate conditions on , and on the kernel .
Daewook [7] studied the following viscoelastic Kirchhoff equation with nonlinear source term and varying time delay:which is a description of axially moving viscoelastic materials. According to the smallness condition taking into account of Kirchhoff coefficient and the relaxation function and by summing if or if , he got the uniform decay rate of the Kirchhoff type energy.
Very recently, in [1], we have proved the global existence and energy decay of solutions of the following viscoelastic nondegenerate Kirchhoff equation:with respect to some proposed assumptions. Under assumption setting on , , , and , the authors have obtained the global existence of solution and the decay rate of energy.
Recently, Mezouar and Boulaaras [1] have studied viscoelastic nondegenerate Kirchhoff equation with varying delay term in the internal feedback.
In the present paper, we extent our recent published paper in [1] for a coupled system (3). The famous technique using the presence of delay in PDE’s problem is to set a new variable defined by velocity depending on delay, which will give us new problem equivalent to our studied problem but the last one is a coupled system without delay. After this, we can prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Faedo–Galerkin procedure, and under a choice of a suitable Lyapunov functional, we establish an exponential decay result.
The outline of the paper is as follows. In Section 2, some hypotheses related to problem are given and we state our main result. Then, in Section 3, the global existence of weak solutions is proven. Finally, in Section 4, we give the uniform energy decay.
1.2. Preliminaries and Assumptions
We denote by (.,.) the inner product in .
Now, we introduce, as in [8], the new variables
Then, we have
Similarly, we have
Therefore, problem (3) is equivalent to
To state and prove our result, we need some assumptions.
Assumption 1. Assume that satisfies
Assumption 2. For the relaxation functions, are bounded functions such thatand suppose that there exist positive constants satisfyingfor .
Assumption 3. is a function in , , such thatWe define the energy associated to the solution of system (7) bywhere is a positive constant such that
Theorem 1 (global existence). Let , s, and satisfy the compatibility conditionAssume that Assumptions 1–3 hold. Then, problem (3) admits a weak solution such that , , and .
Theorem 2 (decay rates of energy). Assume that Assumptions 1–3 hold. Then, for every there exist positive constants and such that the energy defined by (12) possesses the following decay:
2. Preliminaries
Lemma 1 . Let q be a number withThen, there exists a constant such that
The following lemma states an important property of the convolution operator.
Lemma 2 (see [9]). For , we have
Lemma 3. Let be a solution of problem (7). Then, the energy functional defined by (12) satisfieswhere , , and are positive.
Proof. Multiplying the first equation in (7) by , integrating over , and using integration by parts, we getConsequently, by applying Lemma 2, equation (21) becomesSimilarly by multiplying the second equation in (7) by , integrating over , and using integration by parts, we getMultiply the third equation in (7) by and integrate the result over to obtainConsequently,Similarly, we getCombining (22)–(26), we obtainFrom Assumption 3, we getUsing (13), this completes the proof.
3. Global Existence (Proof of Theorem 1)
Throughout this section we assume , and . We will use the Faedo–Galerkin method to prove the existence of global solutions. Let be fixed and let be a basis of and be the space generated by . Now, we define, for , the sequence as follows:
Then, we may extend by over such that forms a basis of and denote the space generated by . We construct approximate solutions , , in the formwhere , , , and are determined by the following ordinary differential equations:
Noting that , from the generalized Hölder inequality, we obtain
Since Assumption 2 holds, according to Sobolev embedding, the nonlinear terms and in (31) make sense.
The standard theory of ODE guarantees that systems (31)–(38) have a unique solution in , with . In the next step, we obtain a priori estimates for the solution of systems (31)–(38), so that it can be extended outside to obtain one solution defined for all , using a standard compactness argument for the limiting procedure.
3.1. First Estimate
Since the sequences , and converge and from Lemma 3 with employing Gronwall’s lemma, we can find a positive constant independent of such thatwhere
Noting Assumption 1 and estimate (40) yields that
3.2. Second Estimate
Multiplying the first equation (respectively, the second equation) in (31) by (respectively, by ) and summing over j from 1 to k, it follows that
Differentiating (36) with respect to t, we get
Multiplying the first equation by (respectively, the second equation by ) and summing over j from 1 to k, it follows thatand then we have
Integrating over with respect to , we obtain
Summing (43) and (47) and as , we get
By Young’s inequality, the right hand side of (48) can be estimated as follows:
Using Cauchy–Schwarz inequality and Sobolev–Poincare inequality, we obtain
Similarly,
By using (49)–(53) in (48), we deduce
By using Assumption 3 and taking the first estimate (40) into account, we inferwhere is a positive constant which depends on .
Integrating (55) over (0, t), we obtain
For a suitable such that for and using Gronwall’s lemma, we obtain the second estimate
We observe from estimates (40) and (57) that there exist subsequences of and of such that
In the following, we will treat the nonlinear term. From the first estimate (40) and Lemma 1, we deduce
On the other hand, from Aubin–Lions theorem (see [10]), we deduce that there exists a subsequence of , still denoted by such thatwhich implies
Hence,where . Thus, using (64) and (66) and Lions lemma, we derive
Similarly,which implies almost everywhere in .
By multiplying (31) and (36) by and by integrating over , it follows thatfor all .
The convergence condition in (58)–(62), (68) and (67) are sufficient, thus we can pass to the limit in (70). Then, we havefor all . This completes the proof of Theorem 1.
4. Uniform Decay of the Energy (Proof of Theorem 2)
In this section we study the solution’s asymptotic behavior of system (3).
To prove our main result, we construct a Lyapunov functional equivalent to . For this, we define some functionals which allow us to obtain the desired estimate.
Lemma 4. Let be a solution of problem (7). Then, the functionalsatisfies the estimate.(i).(ii).
Proof. (ii) Differentiating (72) with respect to t and using (5)-(6) and Assumption 3, we getSince is a decreasing function for and , we deduceThus, our proof is completed.
Lemma 5. Let be a solution of problem (7). Then, the functionalsatisfies the estimate(i).(ii).
Proof. (i)From Young’s inequality, the Sobolev embedding, and , we deduce(i)Differentiating with respect to t and using the first and second equations of (7), we getAs and making use Young’s inequality, we obtainBy use of Young’s inequality, we can estimate the third term in the right side as follows:Similarly,Thus, our proof is completed.
Lemma 6. Let (u, z) be a solution of problem (7). Then, the functionalsatisfies the estimates(i).(ii),where and is the Sobolev embedding constant.
Proof. (i). We use Young’s inequality with the conjugate exponents and , and the second term in the right hand side can be estimated as We have by Hölder’s inequality Combining (83) with (82), we obtain Similarly, we get Similarly, Combining (84)–(86), we deduce (i).(ii)Using the Leibniz formula and the first and second equations of (7), we havewhereIn what follows we will estimate .
For , we use Hölder and Young’s inequalities with , and we getSimilarly,and since , we follow almost the same steps to obtain )Combining (87) and (89)–(94), we finish the proof.
Proof of Theorem 1.2. Now, for , we introduce the following functional:Indeed, to prove , we show that there exist two positive constants and such thatFrom (i) of Lemmas 4–6 and recalling the fact that , we getwhere depending on . If we choose , we can obtain our result.
By recalling Lemmas 3–6 and by Assumption 2, we deduce that for ,where , , , , and .
Let be sufficiently small so is fixed; we take and small enough such thatFurther, we choose small enough such thatThus,where and .
Let . From (101), we getfor some . A simple integration over yieldsThanks to equivalence between and , we obtain (16).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally to this study. All authors read and approved the final manuscript.