Global Existence and Energy Decay Rates for a Kirchhoff-Type Wave Equation with Nonlinear Dissipation

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 Ku′′ + M(|A 1/2 u|2)Au + g(u′) = 0 under suitable assumptions on K, A, M(·), and g(·). Next, we derive decay estimates of the energy under some growth conditions on the nonlinear dissipation g. Lastly, numerical simulations in order to verify the analytical results are given.


Introduction
A mathematical model for the transverse deflection of an elastic string of length > 0 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 > 0, 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 Kirchhofftype model also appeared in scientific research for beam or plate [2][3][4][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][7][8][9][10][11][12][13][14].
Equation (2) with linear dissipative term, that is, ( ) = ( > 0), 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 2 The Scientific World Journal dissipative term. But they studied a simple form with the coefficient (⋅) ≡ 1. 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 (⋅) ∈ 1 [0, ∞).
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.

Preliminaries and Existence
Let Ω be a bounded open domain in R having a smooth boundary Γ and = 2 (Ω) 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 > 0 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 := ( 1/2 ). Throughout the paper we will make the following assumptions. (G) : R → R is a nondecreasing continuous function such that (0) = 0 and there is a constant > 0 and ≥ 1 such that And ( ( ), ) ≥ 0 for all ∈ ( ) ∩ ( 1/2 ). Note that the last assumption of (G) makes sense. In fact, when = −Δ and ( ) = | | , ≥ 1, we can easily show that ( ( ), ) ≥ 0 for all ∈ ( ) ∩ ( 1/2 ).
Let ( ) and ( ) be defined as follows: And also let us consider the functions Theorem 1. Let the initial conditions ( 0 , 1 ) ∈ × 2 (Ω) satisfy the smallness assumption ; ) such that, for any > 0, Proof. Assume that, for simplicity, is separable; then there is a sequence ( ) ≥1 consisting of eigenfunctions of the operator corresponding to positive real eigenvalues tending to +∞ so that = , ≥ 1. Let us denote by the linear hull of 1 , 2 , . . . , . Note that ( ) ≥1 is a basis of , , and and hence it is dense in , , and .
The Scientific World Journal 3 Approximate Solutions. We search for a function ( ) = ∑ =1 ( ) such that, for any V ∈ , ( ) satisfies the approximate equation and the initial conditions as the projections of 0 and 1 over satisfy For V = , = 1, 2, . . . , the system (13)-(15) of ordinary differential equations of variable has a solution ( ) in an interval [0, ).
Now we obtain a priori estimates for the solution ( ) and it can also be extended to [0, ) for all > 0.
A Priori Estimate I. Let us consider V = in (13). Using (7), we have Integrating (16) over (0, ), ≤ , 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 0 < < ∞, Now we show that ( ) can be extended to [0, ∞). We need the following smallness assumption: With simple computations it follows that for all ∈ [0, ). Next, we show that = * . Let us assume by contradiction that < * . Since | ( 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 [0, ) for any ∈ (0, ∞).
Furthermore, putting V = in (13), we get From this we obtain Integrating (34) From (6) and (22), it follows that A Priori Estimate II. Taking V = ( ) in (13) and choosing = 0, 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 0 < < − . By taking the difference V = ( + ) − ( ) in (13) and the assumption (G), we obtain Thus we have Set The Scientific World Journal 5 By using (42), Young's inequality, the assumption (M), and the fact that is positive self-adjoint operator, we see that Φ ( ) ≤ Φ ( ). Therefore we deduce Dividing the two sides of (44) by 2 , letting → 0, and using (43), we deduce From (40), it follows that | | 2 ≤ . Since ∈ 2 [0, ], the previous inequality is verified for all ∈ [0, ]. Therefore we conclude that bounded in ∞ (0, ; ) .

Energy Estimates
In this section we study the energy estimate under suitable growth conditions on .
Let us assume that there exist a number ≥ 1 and positive constants 1 , = 1, 2, such that for all ∈ R. Theorem 3. Assume that (65) holds. Then one obtains the following energy decay: where 0 , , and̃0 are some positive constants.
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 ( ) ( −1)/2 and integrating by parts, we have Note that by the assumption (M) and (21), we can choose some positive number so that 2 ( ) ≤ | 1/2 | 2 + | 1/2 | 2 . 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 3 of (72), we set The Scientific World Journal 7 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 > 0, It remains to estimate the second term of 3 . 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 > 0, Combining (80) with (85) and setting̃= 1 + −1 , we obtain Therefore we conclude that Now we choose as ∈ (0, 2 /(3 + 1) ); then (69) follows.

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Numerical Result
In this section, we consider a Kirchhoff-type equation with heterogeneous string as an application: in ( , ) ∈ (0, 1) × (0, 3) , where is a positive constant and ( ), are given in Table 1.
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 = 3 when ( ) = | | 2 . Therefore, by Theorem 3, we get the energy decay rates for the energy ( ) as follows.
Theorem 5. We obtain the following energy decay: where 1 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]). In case of = 10 and = 10 −0.3 , 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 1 ( +1) 1 and ( ) per the two values, respectively, where the parameter value 1 = 30.2 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 | | 2 .