The Stability of Solutions for the Generalized Degasperis-Procesi Equation with Variable Coefficients

Abstract

A generalized Degasperis-Procesi equation with variable coefficients is investigated. The stability of the strong solution for the equation is established under certain assumptions.

1. Introduction

The Degasperis-Procesi (DP) equation was discovered by Degasperis and Procesi [1] in a search for integrable equations similar to the Camassa-Holm equation. Degasperis and Procesi [1] studied a family of third order dispersive nonlinear equations where . It is found in [1] that there are only three equations that satisfy asymptotic integrability conditions within this family. By rescaling and applying a Galilean transformation, the three equations are Korteweg-de Vries equation the Camassa-Holm equation and the Degasperis-Procesi equation (1). Degasperis et al. [2] proved the formal integrability of (1) and the existence of the nonsmooth solutions by constructing a Lax pair.

In recent years, (1) which plays a similar role in water wave theory as the Camassa-Holm equation has caused extensive concern of many scholars (see [1–11]). For example, Coclite and Karlsen [3] established the well-posedness of weak solutions for (1). They proved uniqueness within a class of discontinuous solutions to (1) in [4]. Escher et al. [5] established the precise blow-up rate and proved the existence and uniqueness of global weak solutions to (1) in which the initial data satisfied appropriate conditions. Lai and Wu [7] investigated the local well-posedness of solutions to a generalization of both (1) and (4) in the Sobolev space with . Lenells [8] classified all weak traveling wave solutions of the Degasperis-Procesi equation (1). Ai and Gui [9] proved global existence of solutions for the viscous Degasperis-Procesi equation and showed that the blow-up phenomena occurs in finite time. Fu et al. [11] studied the orbital stability of the peakons for the Degasperis-Procesi equation with a strong dispersive term on the line and proved that the shapes of these peakons were stable under small perturbations.

As we know, their coefficients play an important role to study the fundamental dynamical properties of the Degasperis-Procesi models. It prompts us to study the following generalized Degasperis-Procesi equation: where , , and function is a polynomial of order . Letting , , , (5) reduces to the Degasperis-Procesi equation (1). We consider the Cauchy problem of (5) with an initial condition . Namely, Assume that (5) possesses a bounded strong solution in its maximum existence time interval and lies in . We use the approaches of Kružkov doubling the variables presented in [12] to prove the stability of the solution for the variable coefficients equation (5). From our knowledge, it has not been acquired in the literature.

This paper is organized as follows. Section 2 gives several lemmas. The proof of local solution stability is presented in Section 3.

2. Preliminaries

Applying the operator to (5), we obtain its equivalent form where .

Equations (5) and (6) are equivalent to the problem where and . Notice that .

Remark. According to the statements presented in [7] or [12], we know that problem (8) has a unique local solution in the space if we assume .

Assume that and are solutions of problem (8) in the domain with initial functions and , where is the maximum existence time of solutions. For simplicity, we denote by any positive constants. Now we give several lemmas.

Lemma 1. Let be the solution of problem (8) and . Then where positive constant depends on and .

Proof. We have in which we have used to complete the proof.

Lemma 2. Assume that and are solutions of problem (8) in the domain , , and . Then where depends on and .

Proof. Using the property of the operator , we get in which we apply the Tonelli Theorem to complete the proof.

Let , , for ; and is infinitely differential on . Set , where is an arbitrary positive constant. It is found that and

Let the function be defined and locally integrable on . Set ; denote the approximation function of as We call a Lebesgue point of the function if At any Lebesgue point , we get Since the set of points which are not Lebesgue points of has measure zero, we have as almost everywhere.

For any , we denote the band by . Let and where , .

We state the concept of a characteristic cone. Let , for any ; we define Let represent the cone , and let designate the cross section of the cone by the plane .

Lemma 3 (see [12]). Let the function be bounded and measurable in cylinder . For any and any , the function satisfies

Lemma 4 (see [12]). If is bounded, the function satisfies the Lipschitz condition in and .

Lemma 5. If is the solution of problem (11) on , , it holds that where is an arbitrary constant.

Proof. Suppose that is a twice differential function. Multiplying the first equation of problem (8) by and integrating over , we get Using the method of integration by parts, we get Notice that So Then we have Substituting (23) and (26) into (22), we get Let be an approximation of the function . When , . Setting , then , . Hence, combining with (27), we complete the proof.

3. Main Result

Set function , , outside the cylinder , where , , . Now we give the main result of this work.

Theorem 6. Assume that and are two strong solutions of problem (8) with initial data (). Let be the maximum existence time of and . If and , for any , it holds that where is a positive constant depending on and .

Proof. Set , . Using the Kružkov device of doubling the variables in [12] and Lemma 5, we get Similarly, we have Adding (31) and (32), we obtain Set function in which and . Thus, we obtain We will prove that the form in (33) approaches zero as . In fact, the coefficients of and in vanish for . Thus the integrals of can be rewritten as the following concrete form: In the following computations, we omit the index of function . Applying the Taylor formula, we have the relations It is seen that the identity . In a similar way, we obtain The functions , , and in (37) and (38) satisfy where and as . There are for or and Hence, we get where as .
We denote the integrand in (41) as where and satisfy the Lipschitz condition in . Applying the property of the function and the method of integration by parts, we have Hence Using Lemma 3, as . Therefore, we have It follows from (33) to (45) that We note that the first two terms of the integrand of (46) have the form where satisfies the Lipschitz condition in all its variables. Then Note that outside the region . Applying the estimate and Lemma 4, we get where is a positive constant independent of . Using Lemma 3, we know as .
For the term , we substitute , , , . Combining with the identity we obtain Thus, we have Similarly, the integrand of the third term in (46) can be represented as Then It holds that Using Lemma 3, it yields as . Repeating the steps as before, we have From (46) to (56), we get We set Take two numbers , and . In (57), we set in which where is a small positive constant and outside the cone . When , we observe that . By the definition of the number , we have Applying (57)–(60), we get In (62), sending and using Lemma 2, we obtain where is independent of .
Applying the properties of the function for , we get Then Let We observe that Letting , it derives that Therefore, we have From (63)–(69), we obtain inequality Choosing , we get Applying the Gronwall inequality, we complete the proof of Theorem 6.

Conflict of Interests

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

Acknowledgment

This work is supported by National Natural Science Foundation of China (11471263).