A Semilinear Wave Equation with a Boundary Condition of Many-Point Type: Global Existence and Stability of Weak Solutions

Abstract

This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.

1. Introduction

Recently, initial-boundary value problems of wave equations have appeared more and more in mechanics, they have been deeply studied by many authors, and we can refer to the works [1–13]. In this paper, we consider the following nonlinear wave equation with a boundary condition of many-point type:where are given real functions and are positive constants such that .

Dang and Alain [4] studied the global existence of the following problem:where is a constant and are given functions.

In [11], Santos considered the following problem:in which are given functions. He studied the asymptotic behavior of the solutions of problem (3) with respect to the time variable. In this case, problem (3) is a mathematical model for a linear one-dimensional motion of an elastic bar connected with a viscoelastic element at an end of the bar.

Applying the Mikusinski operational calculus, D. Takači and A. Takači [12] gave the formula for finding the exact solutions of the linear wave equation given bywhere are given nonnegative constants and are given functions. Also, the unknown function and the unknown boundary value satisfy the following integral equation:It is worth noting that the function is deduced from a Cauchy problem for an ordinary differential equation at the boundary condition . Indeed, if satisfies the following Cauchy problemwhere are constants such that , we can easily show thatwherewith . In the special case of , we obtain (5).

Besides, Nguyen and Giang Vo [8] obtained the asymptotic behavior of the weak solution of the following initial-boundary value problem as :in which are given constants and are given functions. Problem (9) is said to be a mathematical model describing a shock problem involving a linear viscoelastic bar.

The organization of this paper is as follows. First of all, we establish the global existence and uniqueness of weak solutions of problem (1). The proof is based on the Galerkin method associated with a priori estimates and the weak compact method. Finally, here we prove that this solution is stable in the sense of continuous dependence on the given data . This paper is a relative generalization of the works [4, 6, 11, 12].

2. Preliminaries

For convenience, we denote by and , respectively, the scalar product and the norm in . Also, we define a closed subspace of the Sobolev space as follows:with the following scalar product and norm:Then it is easy to show the following.

Lemma 1. The embedding is compact and

On the other hand, we also have the following result.

Lemma 2. Let . Then

The proof of the lemma is straightforward; we omit the details.

Remark 3. Let , we get

Next, if is a real Banach space with norm , consists of equivalence classes of strongly measurable functions such that , withIt is not difficult to prove that is a Banach space.

Let , , and be Banach spaces satisfying . We further assume that and are reflexive, and the imbedding is compact. Setwhere . Then is a Banach space with the normWe also have the following lemma.

Lemma 4 (see [14]). Let . The embedding is compact.

3. Global Existence and Uniqueness of Weak Solutions

To investigate the existence of a unique weak solution of problem (1), the following assumptions are needed:) and .().() satisfy the following conditions:(i)There exist positive functions such that (ii)There exist positive functions and such that (); there exist constants such that (); there exist constants such that ()For each , there exists a constant such that With these assumptions, we have the following theorem.

Theorem 5. Let assumptions hold. Then problem (1) has a unique weak solution such that

Remark 6. In the special case of and , for all , we have obtained the same results in the paper [6].

Proof of Theorem 5. The main tool of this proof is the Galerkin method. The procedure includes four steps as follows:(i)Galerkin approximation.(ii)A priori estimates.(iii)Limiting process.(iv)Uniqueness of the weak solutions.

Step 1 (Galerkin approximation). We use a special orthonormal base of : Now we are looking for the approximate solution of problem (1) in the formwhere the coefficient functions satisfy the following system of nonlinear differential equationswithBy substituting , we can rewrite the system of (26)–(29) as follows:Therefore, we obtainApplying the Schauder fixed-point theorem, it is not difficult that system (31) has a solution on an interval . This implies that in system (26)–(29) there exists a solution to on . Moreover, we can extend the approximate solution to the whole interval (see [15]).

Step 2 (a priori estimates). In (26), we replace by . Then integrating from to , we have after some calculationswhereWe will estimate, respectively, the following terms on the right-hand side of (32).

Estimating . Using (33) and Lemma 1, we infer thatHence,Estimating . Using Lemma 1 and inequality (34), we arrive atEstimating . From assumption (), we haveOn the other hand, we see thatIt follows from (37), (38), and Lemma 2 thatEstimating . Owing to assumption ()-(i), (38), and Lemma 2, it is not difficult to show thatEstimating . Applying integration by parts, it follows thatWe can estimate the integrals in the right-hand side of (41) as follows:Going in for the Cauchy-Schwartz inequality, we arrive atConsequently,It follows from (44) and (46) thatWe deduce from the estimates of thatwith On the other hand, by (28), (29), and assumptions , where is a positive constant depending only on .

Combining (32), (35), (36), (39), (40), (48), and (50), we obtain after some rearrangementswhereChoosing , by Gronwall’s inequality, we havewhere is a positive constant depending on .

Next, we will require the following lemma.

Lemma 7. There exists a positive constant depending only on such that

Proof of Lemma 7. We putIn view of (25) and (31), can be rewritten as follows:In connection with , we have the following lemma.

Lemma 8. There exists a positive constant depending only on such that

Proof of Lemma 8. We definewithOn the other hand, use the inequality Hence,We will estimate each term on the right-hand side of this inequality.

Estimating . Thanks to (60) and (65), Now, we will need the following lemma.

Lemma 9. Let and . Then

The proof of this lemma is simple; we omit the details.

Applying Lemma 9, we deduce from (28), (66), and assumption () thatwhere always indicates a constant depending on .

Estimating . Similarly, we also obtainEstimating . We see thatTo estimate , we need the following lemma.

Lemma 10. Let . One always has

Proof of Lemma 10. First, we assume that . Set (which is an integer part of ). We consider two cases of .

Case 1 (). ThenSince , for all , we getMoreover, the function is decreasing on ; hence, ; for all , it follows thatOn the other hand, it is easy to verify the following equality by the induction:Using (74) and (75), we arrive atConsequently, it follows from (72), (73), and (76) that

Case 2 (). We haveCombining (77) and (78), we conclude thatSince and the function is even, periodic with the period 2, thus inequality (79) holds for all . The proof of Lemma 10 is complete.

By (70) and Lemma 10, it leads toHence, it follows from (28), (80), and assumptions (), () thatEstimating . Proving in the same way as (81), we getOn the other hand, using assumption ()-(ii), thenwith .

Applying the Cauchy-Schwartz inequality, we clearly getCombining (65), (68), (69), (81), and (84), we obtain Lemma 8.

Remark 11. Lemma   in [4] is a special case of Lemma 8 with and .

We now return to the proof of Lemma 7.

Note that it follows from (27) thatOn account of (53) and assumptions (), (), and (), we getTherefore, it is easy to see thatApplying Lemma 10 and the imbedding , we deduce from (27) and (57) thatThus,Using the Cauchy-Schwartz inequality and Lemma 8, then we obtain from (87) and (89) thatHence,By the Gronwall inequality, we get Lemma 7. This completes the proof of Lemma 7.

Step 3 (limiting process). Due to (53) and (54), applying the Banach-Alaoglu theorem, we can extract a subsequence of sequence , still labeled by the same notations, such thatThanks to Lemma 4 and the compactness of the imbedding , they lead us toBy (93) and assumptions , , and , we havestrongly in .

Also, we apply the inequalitywhere . By (93) and (95), we getPassing to the limit in (26) by (92), (94), and (96), we have satisfying the equationfor all , a.e. .

On the other hand, it is easy to show a similar way as in [4, p. 588]The existence of global solutions is proved.

Step 4 (uniqueness of the weak solutions). Let and be two weak solutions of problem (1). Then is a weak solution of the following problem:in whichTo prove , then the following lemma is needed.

Lemma 12. Let be the weak solution of the following problem:Then we haveEquality holds in case of .
The proof of Lemma 12 is the same as Lemma   in [16].

Applying Lemma 12 with , we getwith .

Now we can estimate the integrals in the right-hand side of (103) as follows.

First term : using assumption (), it follows from Lemma 1 and (103) thatwhereSecond term : set . Integrating by parts, then we arrive atOn the other hand, we easily show thatApplying Lemma 2, we deduce from (107) thatThus, it follows from (106) and (108) thatin whichThird term : using the Cauchy-Schwartz inequality, we have from (103), (), and () thatIn addition, we can similarly prove as in (111) for the fourth and fifth terms as follows:Choosing , the combination of (103), (104), (109), and (111)-(112) shows thatwhereBy the Gronwall inequality, we see that ; that is, .

This completes the proof of Theorem 5.

4. Stability of the Weak Solutions

In this section, let be fixed functions. Also, we assume that , and , are fixed functions, constants satisfying assumptions ()–() (independent of , , and ). Applying Theorem 5, then problem (1) has a unique weak solution depending on . We denotewhere satisfy assumptions ()–().

Then the stability of the solutions of problem (1) is given as follows.

Theorem 13. Let ()–() hold. Then the solutions of problem (1) are stable with respect to the data in the following sense.
If and satisfy the assumptions ()–() such thatstrongly in , as , for all .
Thenwhere .

Proof of Theorem 13. Firstly, we assume thatwhere are fixed positive constants.
On the other hand, by the proof of Theorem 5, the a priori estimates of the sequence satisfywhere is a constant depending only on .
Due to (119) and (92), we conclude thatIn addition, we can prove in a similar way above that the solution of problem (1) corresponding to the data also satisfieswith being a constant depending only on .
We setThen satisfies the following problem:whereApplying Lemma 12 with , we see thatwhereLet . Now we can estimate eight integrals in the right-hand side of (125) as follows.
Estimating . From assumption (), it yieldswith .
Estimating . It is easy to show thatEstimating . Since , by Cauchy-Schwartz inequality, thenMoreover, using the imbedding , there exists a positive constant such thatThus,Estimating . By help of assumption () and (121), we getEstimating . Proving in a similar way to (109), we also obtainwhereOn the other hand, we haveTherefore,It impliesEstimating . By reusing the inequalities (121) and (135) and assumption (), thenEstimating . Similarly, from assumption (), we also obtainEstimating . Due to assumptions () and (), we getwhereThus, combining (125)–(128), (131), (132), and (137)–(140), in whichChoosing and applying the Gronwall inequality, we deduce from (120), (121), and (142) thatThis shows thatwith being a constant depending only on .
This completes the proof of Theorem 13.

Remark 14. If we use the inequality , then, with regard to (126) and (145), conclusion (117) in Theorem 13 can be extended as follows:strongly in ,  as  .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.