Global Existence of Solutions for the Viscoelastic Kirchhoff Equation with Logarithmic Source Terms

Mezouar, Nadia; Boulaaras, Salah Mahmoud; Allahem, Ali

doi:https://doi.org/10.1155/2020/7105387

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Volume 2020 | Article ID 7105387 | https://doi.org/10.1155/2020/7105387

Global Existence of Solutions for the Viscoelastic Kirchhoff Equation with Logarithmic Source Terms

Nadia Mezouar,¹Salah Mahmoud Boulaaras,^2,3and Ali Allahem⁴

Guest Editor: Karthikeyan Rajagopal

Received24 Jan 2020

Revised16 Feb 2020

Accepted27 Feb 2020

Published04 Apr 2020

Abstract

In this paper, a nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay term and logarithmic nonlinearity in the weakly nonlinear internal feedback is considered, where the global and local existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo-Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given. These results extend and improve many results in the literature.

1. Introduction

1.1. Model

In this paper, we consider the global existence and decay properties of solutions for the initial boundary value problem of the following viscoelastic nondegenerate Kirchhoff equation of the form:where is a bounded domain in , , with a smooth boundary , , , , and are positive real numbers, is a positive function which decays exponentially, is a time-varying delay, and are two functions, and the initial data are in a suitable function space. is a -function for , with and .

In the absence of delay term (i.e., ), Han and Wang in [1] considered the following nonlinear viscoelastic equation with damping:

Time delay is often present in applications and practical problems. In recent years, the control of PDEs with time delay effects has become an active area of research (see, for example, [2–4]). For example, in [5], it has been proven that a small delay in a boundary control could turn a well-behaved hyperbolic system into a wild one, thus showing that delay can be a source of instability.

Wu [6] treated problem (1) for a constant time delay and . He proved the local existence result using the Faedo-Galerkin method and established the decay result employing suitable Lyapunov functionals under appropriate conditions on and and on the kernel .

Benaissa et al. [7] considered the case of constant time delay , with and . They proved the global existence and uniform decay for the following problem:

The same problem (3) was also treated by Kirane and Said-Houari [8] for and a homogeneous right hand side with , a constant time delay. Daewook [9] considered a viscoelastic Kirchhoff equation, with a time-varying delay and a nonlinear source term, given as

This equation describes axially moving viscoelastic materials. Using the smallness condition with respect to Kirchhoff coefficient and the relaxation function and by assuming if or if , he obtained the uniform decay rate of the Kirchhoff-type energy.

In [10], the authors studied homogeneous problem (1) without the viscoelastic term, with and . In addition, and are multiplied by a positive nonincreasing function of satisfying and . They proved the global existence, and using a multiplier method with some properties of convex functions to get decay rate of the energy (when goes to infinity) depends on the function and on the function which represents the growth at the origin of .

Apart from the aforesaid attention given to polynomial nonlinear terms, logarithmic nonlinearity has also received a great deal of interest from both physicists and mathematicians. This type of nonlinearity was introduced in the nonrelativistic wave equations describing spinning particles moving in an external electromagnetic field and also in the relativistic wave equation for spinless particles [11]. Moreover, the logarithmic nonlinearity appears in several branches of physics such as inflationary cosmology [12], nuclear physics [13], optics [14], and geophysics [15]. With all this specific underlying meaning in physics, the global-in-time well-posedness of solution to the problem of evolution equation with such logarithmic-type nonlinearity captures lots of attention. Birula and Mycielski [16, 17] studied the following problem:which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit goes to 1 for the -adic string equation [18, 19]. In [20], Cazenave and Haraux consideredand they established the existence and uniqueness of the solution for the Cauchy problem. Gorka [21] used some compactness arguments and obtained the global existence of weak solutions, for allto initial boundary value problem (5) in the one-dimensional case. Bartkowski and Górka [22] proved the existence of classical solutions and investigated the weak solutions for the corresponding one-dimensional Cauchy problem for equation (6). Hiramatsu et al. [23] introduced the following equation:to study the dynamics of Q-ball in theoretical physics and presented a numerical study. However, there was no theoretical analysis for the problem. In [24], Han proved the global existence of weak solutions, for allto initial boundary value problem (8) in .

In the present paper, we investigate the stabilization of a dynamic model describing a string with a rigid surface and an interior somehow permissive to slight deformations. This leads to a varying material density and a Kirchhoff term that depends on . We prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Fadeo-Galerkin procedure. We also establish an explicit and general decay result using a perturbed energy method with some techniques due to Mustafa and Messaoudi [25], as well as some properties of convex functions. These convexity arguments were introduced and developed by Lasiecka et al. [26–28] and used, with appropriate modifications, by Liu and Zuazua [29], Alabau-Boussouira [30], and others.

The paper is organized as follows: In Section 2, we give some hypotheses and state our main result. Then, in Section 3, we prove the global existence of weak solutions. Furthermore, in Section 4, the uniform decay of the energy is derived.

1.2. Formulation of the Results

We denote by (., .) the inner product in and the corresponding norm by . Now, we introduce, as in [31], the new variable:

Then, we have

Therefore, problem (1) is equivalent to

To state and prove our result, we need some assumptions. (A1) Assume that satisfies (A2) The relaxation function is a bounded function, such that and suppose that there exists a positive constant satisfying (A3) is a nondecreasing function of class and is convex, increasing and of class , satisfying where are positive constants. is an odd nondecreasing function of class such that there exist , (A4) is a function in , , such that where and are positive numbers. (A5) We also assume that (A6) We define the energy associated to the solution of system (12) by where is a positive constant such that

Theorem 1. (Global Existence). Let satisfy the compatibility condition:Assume that (A1)–(A6) hold under smallness condition on the initial data . Then, problem (1) admits a weak solution:

Theorem 2. (Uniform Decay Rates of Energy). Assume that (A1)–(A6) hold and if is positive and bounded, then for every , there exist positive constants , and such that the solution energy of (1) satisfieswhere

Here, is strictly decreasing and convex on with .

2. Preliminaries

Lemma 1. (Sobolev–Poincaré’s Inequality). Let be a number withThen, there exists a constant such that

Lemma 2. (see [32, 33]) (Logarithmic Sobolev Inequality). Let be any function in and be any number. Then,

Lemma 3. (see [20]) (Logarithmic Gronwall Inequality). Let and and assume that the function satisfiesThen,

Lemma 4. Let . Then, there exists such that

Proof. Let . Notice that is continuous on , and its limit at is and its limit at is . Then, has a maximum on , so the proof is complete.
The following lemma states an important property of the convolution operator.

Lemma 5. (see [34]). For , we have

Remark 1. Let us denote by the conjugate function of the differentiable convex function , i.e.,Then, is the Legendre transform of , which is given by (see Arnold [35], p. 61-62)and satisfies the generalized Young inequality:

Lemma 6. Let be a solution of problem (12). Then, the energy functional defined by (22) satisfieswhere and .

Proof. Multiplying the first equation in (12) by , integrating over , and using integration by parts, we getConsequently, by applying Lemma 5, equation (40) becomesMultiplying the second equation in (12) by and integrating the result over , we obtainConsequently,Combining (41) and (43), we obtainFrom (18) and , we getUsing (18) and Remark 1, we obtainHence,Using (18) and (38) with and , we have from (45) thatThis completes the proof.

3. Proof of Theorem 1

3.1. Local Existence

Throughout this section, we assume and . We will use the Faedo-Galerkin method to prove the existence of a solution to problem (1). Let be fixed and let , be a basis of , and let 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 as 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 holds, according to Sobolev, embedding the nonlinear term in (51) makes sense.

The standard theory of ODE guarantees that systems (51)–(55) have a unique solution in , with , by Zorn lemma since the nonlinear terms in (51) are locally Lipschitz continuous. Note that is of class . In the next step, we obtain a priori estimate for the solution of systems (51)–(55), so that it can be extended to and that the local solution is uniformly bounded independently of and .

3.1.1. The First Estimate

Since the sequences , and converge and from Lemma 6, we can find a positive constant independent of such that

As is a positive nonincreasing function, we getwhere

By applying the Logarithmic Sobolev inequality, (58) yieldswhere is a positive constant depending only on , and .

By choosingwe obtain and .

This selection is possible thanks to . So we get

Let us note that

Then, by using Cauchy Schwarz’s inequality, we getHence, (62) giveswhere . Applying the Logarithmic Gronwall inequality to (65), we obtain

Hence, from (58), we obtain the first estimate:

The estimate implies that the solution exists in and it yields

3.1.2. The Second Estimate

Replacing by in (51), multiplying by , and summing over from to , it follows that

Noting that and by using Lemma 5, we obtain

By using the Green formula, we have

Consequently, equation (74) yields

To estimate the term on the right-hand side of (76), we apply Lemma 4 with and use repeatedly Young’s, Cauchy-Schwartz’s, and the embedding inequalities as follows:

Combining (76) and (77) to have

Replacing by in (54), multiplying by , and summing over from to , it follows that

Then, we get

We integrate over , and we find

Combining (78) and (81) and using (A2), we get

From the first estimate (67) and Young’s inequality, we get

Using (17) and Chaucy-Schwarz’s inequality, we obtain

Taking into account (83) and (84) into (82) yields

Multiplying (51) by and summing over from to , it follows that

Then,

Differentiating (54) with respect to , we get

Multiplying by and summing over from to , it follows thatThen, we have

Integrating over with respect to , we obtain

Summing (87) and (91), we get

By Cauchy-Schwarz’s, Sobolev’s, and Young’s inequalities, the right hand side of (92) can be estimated as follows:and from (16),

Using Lemma 6, Jensen’s inequality, and the concavity of , we obtain

From (17) (that is, ), we get

Similar to (77), we get

Substituting (93)–(98) into (92) yields

Combining (85) and (99), we get

Then, from (67) and by integration over , (100) yields

For a suitable , we get

Using Gronwall lemma, we obtain

We observe from the estimates (67) and (103) that there exists a subsequence of and functions such that

Now, we will prove that is the solution of (1). First, we will treat the nonlinear terms.(1)Term : from the first estimate (67) and Lemma 1, we deduce

On the other hand, from Aubin-Lions theorem (see Lions [36]), we deduce that there exists a subsequence of , still denoted by such thatwhich implies that

Hence,where . Thus, using (117), (114), and Lions Lemma, we derivewhich implies almost everywhere in .

(2) Term : using (103), we have being bounded in which implies the boundedness of in . Similarly, is bounded in . Then, from Aubin-Lions theorem, we find a subsequence such thatwhich implies

Since the map is continuous, we have the following convergence:

Using the embedding of in , it is clear that is bounded in . Next, taking into account the Lebesgue bounded convergence theorem , we get

Lemma 7. For each , and , where is a constant independent of .

Proof. By (A2) and (118), we haveHence, by (71) and Fatou’s Lemma, we haveBy using Cauchy-Schwarz’s inequality, (96), and (122), we have

Lemma 8. We have and .

Proof. Let and setwhere is the measure of . If ,By applying (71), we deduce that as . From Vitali’s convergence theorem, we deduce thatHence,Similarly, we have

Remark 2. By using (103) and from (104) and (105) combined with the Aubin-Lions compactness lemma, we deduceBy multiplying (51) and (54) by and by integrating over , it follows thatfor all .
The convergence of (104)–(110), (115), and (124)–(129) is sufficient to pass to the limit in (130) in order to obtainThen, problem (1) has a weak solution on , .

3.2. Global Existence

To state and prove our global existence, we introduce the following functionals:

We note that

Lemma 9. The following inequalities hold:where , is the Lesbegue measure of , and is the smallest embedding constant:

Proof. Let and . By using (136), we haveOn the other hand, using Hölder’s inequality and (136), we getHence, (135) is obtained.

Lemma 10. Assume that (A1)–(A6) hold. Let such thatThen,

Proof. Since and is continuous on , there exists such that , for all . Let us denote by the largest real number in such that on . We assume by contradiction that , so we have and from (133), we haveThe last inequality is obtained from Lemma 6. If , then (132) and (135) giveConsequently, if on , we getThen,which is not true since on . If there exists in such that , then let be the smallest real number such that . Because and is positive, nonincreasing, and continuous on , then and on . Therefore, from (142), we deduce thatThen,As given above, we get a contradiction with the fact that on . Then, we conclude that . From (132), we haveBy using (135) and (141), we haveBy recalling (139), we arrive at , which contradicts the assumption that . Hence, and then on .
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 (1).

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 11. Let be a solution of problem (12). Then, the functionalsatisfies the estimates

Proof. (ii) Differentiating (149) with respect to and using (16), (11), and , we getSince is a decreasing function for and , we deduceThus, our proof is completed.

Lemma 12. Let be a solution of problem (12). Then, the functionalsatisfies the estimateswhere and is the Sobolev embedding constant.

Proof. (i)From Young’s inequality, Sobolev embedding, and Lemma 6, we deduce(ii)Differentiating with respect to and using the first equation of (12), we getBy using Young’s inequality and Sobolev embedding, we can estimate the third term in the right side as follows:Thus, our proof is completed.

Lemma 13. Let be a solution of problem (12). Then, the functionalsatisfies the estimateswhereand is the Sobolev embedding constant.

Proof. (i)We have We use Young’s and Hölder’s inequalities with the conjugate exponents and ; the second term in the right hand side can be estimated as We get the last inequality from (22) and Lemma 6. Similarly, we use Young’s and Hölder’s inequalities with , we get Combining (162) and (163), we deduce(ii)We use the Leibniz formula and the first equation of (12), and we haveIn what follows, we will estimate . For , we use Hölder’s and Young’s inequalities with , and we getwhere obtained by recalling (22) and Lemma 6. Similarly,To estimate , we apply Lemma 4 with and use repeatedly Young’s, Cauchy-Schwartz’s, and the embedding inequalities, as follows:where . Also,As is a positive decreasing function, and then,where obtained by recalling (22) and Lemma 6. Combining (165) and (166)–(173), we finish the proof. Now, for , we introduce the following functional:

Lemma 14. Let be a solution of problem (12). Assume that (A1)–(A6) hold andThen, satisfies, along the solution and for some positive constants , the following estimate:and .

Proof. By (ii) of Lemmas 11–13 and Lemma 6 and by (A2), we deduce that for :whereWe take and sufficiently small such thatAs long as and are fixed, we choose large enough such thatThus,whereUsing the Logarithmic Sobolev inequality, we getFrom (61) and forwe haveThis selection is possible thanks to (A6). So we getwhere . By recalling that and and using (133), (134), and (175), we obtainTaking satisfies(So (61) is satisfied), and we guaranteetwhich completes the proof of (176). To prove , we show that there exist two positive constants and such thatFrom (i) of Lemmas 11–13, (140), (133), and (134), we get depending on such thatFor a choice of large enough such that and , we get our result. By the proof of Theorem 2 as given by Komornik [37], we consider the following partition of :We use Young’s inequality (with ), (22), and Lemma 6, and we haveSimilarly and by application of (17), we obtainCombining (193) and (194), (176) becomeswhere . Now, for small enough such that , the function satisfies

Case 1. is linear on ; using (16) and Lemma 6, we deduce thatThus, satisfiesHence,

Case 2. is nonlinear on ; so we exploit Jensen’s inequality (see [2]) and the concavity of to obtainThen, (196) becomesFor and , we define byThen, we easily see that for ,By recalling that on and using (202), we obtainUsing Remark 1 with , the convex conjugate of in the sense of Young, we obtainwhere is a positive constant depending of . By taking small enough such that and , we obtainwhere a positive increasing function on . By setting , we easily see that, by (204), we haveUsing (207), we arrive atBy recalling (28), we deducewhich givesA simple integration leads toConsequently,Using (208) and (213), we obtain (27). The proof is completed.

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 manuscript.

Authors’ Contributions

All authors contributed equally to this article. They have all read and approved the final manuscript.

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Copyright © 2020 Nadia Mezouar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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