Abstract
The first objective of this paper is to prove the existence and uniqueness of global solutions for a Kirchhoff-type wave equation with nonlinear dissipation of the form under suitable assumptions on , and . Next, we derive decay estimates of the energy under some growth conditions on the nonlinear dissipation . Lastly, numerical simulations in order to verify the analytical results are given.
1. Introduction
A mathematical model for the transverse deflection of an elastic string of length whose ends are held a fixed distance apart is written in the form of the hyperbolic equation which was proposed by Kirchhoff [1], where is the deflection of the point of the string at the time and , are constants. Kirchhoff first introduced (1) in the study of the oscillations of stretched strings and plates, so that (1) is called the wave equation of Kirchhoff type. The Kirchhoff-type model also appeared in scientific research for beam or plate [2–5]. Such nonlinear Kirchhoff model gives one way to describe the dynamics of an axially moving string. In recent years, axially moving string-like continua such as wires, belts, chains, and band saws have been the subject of study of researchers [6–14].
The mathematical aspects of the natural generalization of the model (1) in : under some assumptions on , , have been studied, using different methods, by many authors [6, 8, 15–22].
When and , the problem (2)-(3) was studied by Dickey [16] and Bernstein [15] who considered analytic functions as the initial data (see also Yamada [21] and Ebihara et al. [17]). In case when and , Pohožaev [22] obtained the existence and uniqueness of global solutions for the problem (2)-(3). Lions [20] also formulated Pohožaev's results in an abstract context and obtained better results.
Equation (2) with linear dissipative term, that is, , was investigated by Mizumachi [23], Nishihara and Yamada [24], Park et al. [25], and Jung and Choi [26]. In fact, they studied the existence, uniqueness, and the energy decay rates of solutions for the problem (2)-(3). On the other hand, related works to a Kirchhoff-type equation with instead of can be found in Levine [19]. Jung and Lee [27] got the result for a Kirchhoff-type equation with strong dissipative term. But they studied a simple form with the coefficient . In case of the equation concerning nonlinear Kirchhoff-type coefficient, recently, Kim et al. [8], Ghisi and Gobbino [28], and Aassila and Kaya [29] have studied existence and energy decay rates of global (or local) solutions for the equation. By giving some suitable smallness conditions on the sizes of the initial data, they assured global existence and energy decay rates for the solutions.
In this paper, we study the existence, uniqueness, and the decay estimates of the energy for a class of Kirchhoff-type wave equations in a Hilbert space : where and are linear operators in and . For global existence of this problem, we give some suitable smallness conditions. So, the main contribution of these results is to consider a general model which contains the concrete model (2)-(3) and to improve the results of Kouémou-Patcheu [30] and Jung and Choi [26]. Moreover, as an application, we give some simulation results about solution's shapes and the algebraic decay rate for a Kirchhoff-type wave equation with nonlinear dissipation.
The method applied in this paper is based on the multipliers technique [31], Galerkin's approximate method, and some integral inequalities due to Haraux [32].
This paper is organized as follows. In Section 2, we recall the notation, hypotheses, and some necessary preliminaries and prove the existence and uniqueness of global solutions for the system (4) by employing Feado-Galerkin's techniques under suitable smallness condition. In Section 3, we derive the energy decay rates by using the multiplier technique under suitable growth conditions on . Finally, in Section 4, we give an example and its numerical simulations to illustrate our results.
2. Preliminaries and Existence
Let be a bounded open domain in having a smooth boundary and with inner product and norm denoted by and , respectively. Let be a linear, positive, and self-adjoint operator on ; that is, there is a constant such that Let be a linear, self-adjoint, and positive operator in , with domain dense in , on , and the graph norm denoted by . We assume that the imbedding is compact. Identifying and its dual , it follows that , where is the dual of . Let denote the duality pairing between and and .
Throughout the paper we will make the following assumptions.(M) is a real function and . Furthermore, there exist some positive constants and such that for all and .(G) is a nondecreasing continuous function such that and there is a constant and such that And for all . Note that the last assumption of (G) makes sense. In fact, when and , we can easily show that for all .(H), , .(S) for some .
Let and be defined as follows:
And also let us consider the functions
Theorem 1. Let the initial conditions satisfy the smallness assumption where , − . Then there is a unique function such that, for any ,
Proof. Assume that, for simplicity, is separable; then there is a sequence consisting of eigenfunctions of the operator corresponding to positive real eigenvalues tending to so that , .
Let us denote by the linear hull of . Note that is a basis of , , and and hence it is dense in , , and .
Approximate Solutions. We search for a function such that, for any , satisfies the approximate equation and the initial conditions as the projections of and over satisfy
For , , the system (13)–(15) of ordinary differential equations of variable has a solution in an interval .
Now we obtain a priori estimates for the solution and it can also be extended to for all .
A Priori Estimate I. Let us consider in (13). Using (7), we have Integrating (16) over , , and using (8), we have Using (5) and (7), we deduce that where the left-hand side is constant independent of and . Thus estimation (18) yields, for any ,
Now we show that can be extended to . We need the following smallness assumption: where , / .
Let be the maximal interval where the solution exists. Set and
With simple computations it follows that for all .
Next, we show that . Let us assume by contradiction that . Since in , we have that Since and are continuous functions, by the maximality of we have that necessarily From (88) and (89) it follows that and are nonincreasing functions; hence Moreover by Lemma 3.1 in [28] we have that By (91)–(31), and the smallness assumption (23), we have that
This contradicts (29). Therefore it follows that can be extended to for any .
Furthermore, putting in (13), we get From this we obtain Integrating (34) over and taking into account assumptions (M) and (G), and applying Gronwall’s inequality, we obtain
From (6) and (22), it follows that
A Priori Estimate II. Taking in (13) and choosing , we obtain Thus we have Thanks to the assumption (6), we deduce from (15) that Therefore we conclude that the right-hand side is bounded; that is,
A Priori Estimate III. For , we apply (13) at points and such that . By taking the difference in (13) and the assumption (G), we obtain Thus we have Set By using (42), Young's inequality, the assumption (M), and the fact that is positive self-adjoint operator, we see that . Therefore we deduce Dividing the two sides of (44) by , letting , and using (43), we deduce From (40), it follows that .
Since , the previous inequality is verified for all . Therefore we conclude that Moreover, using (19) and (46), it follows that Applying a compactness theorem given in [33], we obtain
Passage to the Limit. Applying the Dunford-Pettis theorem, we conclude from (19), (21), (36), and (46)-(48), replacing the sequence with a subsequence if needed, that for suitable functions , , and .
Now we are going to show that is a solution of the problem (11)-(12). Indeed, from (49) to (51), we have for each fixed . So we conclude that, for any , which shows that (12) holds.
We will prove that, in fact, ; that is,
For , we have
We deduce from (49) and (54) that the first and second terms in (58) tend to zero as . For the last term, using the fact that is and (21), we can derive (with some positive constants ) Since is bounded in and the injection of in is compact, we have From (58) to (60), we deduce (57). It follows from (49), (51), and (57) that, for each fixed , as .
For the nonlinear term, , it remains to show that, for any fixed , as .
At this moment we use the following lemma due to Jung and Choi (see [26, page 12]).
Lemma 2. Suppose that is a bounded open domain of ; and are in , , such that a.e., in . Then weakly in .
From (53), a.e. in . By (36), we can use the above lemma and so we have ; that is, which implies (62). Therefore we obtain
The uniqueness is obtained by a standard method, so we omit the proof here.
3. Energy Estimates
In this section we study the energy estimate under suitable growth conditions on .
Let us assume that there exist a number and positive constants , , such that for all .
Theorem 3. Assume that (65) holds. Then one obtains the following energy decay: where , , and are some positive constants.
Proof. Let be arbitrary and fixed and let be a solution of (11) and (12). Multiplying (11) by and integrating by parts in , we obtain that
By and being the primitive of an integrable function, it follows that the energy is nonincreasing, locally absolutely continuous and
Here and in what follows we will denote by diverse positive constants. We are going to show that the energy of this solution satisfies
Once (69) is satisfied, the integral inequalities given in Komornik [31] and Haraux [32] will establish (66).
Now, multiplying (11) by and integrating by parts, we have
Note that by the assumption (M) and (21), we can choose some positive number
so that . Thus we deduce that
Using the continuity of the imbedding , the Cauchy-Schwarz and the Young inequalities, we obtain
Hence, since is nonincreasing, we obtain
In order to estimate the last term of (72), we set
Then we have
The Hölder inequality yields
Using (65) and (68), we deduce that
Combining these two inequalities with (77), we obtain
Applying Young's inequality, it follows that, for any ,
It remains to estimate the second term of . Using (88) we have
Similarly, using (6), we obtain
From (81) and (82), we deduce
Using Young's inequality and
it follows from (82) that, for any ,
Combining (80) with (85) and setting , we obtain
Therefore we conclude that
Now we choose as ; then (69) follows.
4. Numerical Result
In this section, we consider a Kirchhoff-type equation with heterogeneous string as an application: where is a positive constant and , are given in Table 1.
Then, the operators , , and the functions and so that we can easily check that the hypotheses (M), (G), (H), and (S) in Preliminaries are satisfied. The smallness condition satisfies . Therefore, by Theorem 1, we can deduce the following results.
Theorem 4. For any , there is a unique solution to the system (88)–(92).
The energy for the system (88)–(92) is given by Next, in order to get the energy decay of (88)–(92), we need the value of the parameter in (65). We can easily check that when .
Therefore, by Theorem 3, we get the energy decay rates for the energy as follows.
Theorem 5. We obtain the following energy decay: where is a positive constant.
For the numerical simulation, we use the finite difference methods (FDM) which are the implicit multistep methods in time and second-order central difference methods for the space derivative in space in numerical algorithms (see [8, 9, 11]).
Figures 1(a)–1(d) show displacements of solutions to the system (88)–(92) with and , respectively.
| (a) Temporal and spatial solution shapes in case of |
| (b) Temporal and spatial solution shapes in case of |
| (c) Temporal and spatial solution contour line in case of |
| (d) Temporal and spatial solution contour line in case of |
In case of and , we deduce the algebraic decay rate for the energy as shown in Figure 2, respectively. The blue line and red dotted circled line (or blue circled line) show and per the two values, respectively, where the parameter value in (94). This result shows that the energy decay rates for solutions are algebraic in case that the system (88)–(92) with the nonlinear damping term .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The first author's research was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. NRF-2013R1A1A2010704). The corresponding author’s research was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. NRF-2012R1A1B3000599).