Abstract
A nonlinear generalized Degasperis-Procesi equation is investigated. Assuming that the strong solution of the equation is bounded in the sense of -norm and the initial data belong to the space , we prove that the solutions are stable in the space .
1. Introduction
Coclite and Karlsen [1] investigated the following generalized Degasperis-Procesi equation: When and satisfies or where is a positive constant, the existence and stability of entropy weak solutions belonging to the class are established for (1) in paper [1].
The objective of this paper is to study the generalized Degasperis-Procesi equation where is a positive constant, is a polynomial of order , and . When and , (4) reduces to the classical Degasperis-Procesi model [2–10]. Assuming that there exists a strong solution to (4), which is bounded in its existence time interval , and the initial value of (4) lies in , we will prove that the strong solutions of the equation are stable in the space (see Theorem 8 in Section 3). From the authors’ knowledge, this is a new result for (4).
This paper is organized as follows. Section 2 gives several lemmas. The main result and its proof are presented in Section 3.
2. Several Lemmas
We consider the Cauchy problem of (4) in the following form: Applying the operator to the first equation of problem (5), we obtain where . Letting , we get
Lemma 1. The solution of problem (5) with satisfies where and . Moreover, there exist two constants and depending only on such that
Proof. Letting and and using (4), we obtain and Using the Parseval identity and (10), we obtain (8) and (9).
Remark 2. When , from (8), we cannot obtain inequality (9).
Lemma 3. If and , it holds that where is a constant independent of and .
Proof. Using the assumption and Lemma 1, we have . Using (7), we get Since the function is a polynomial of order and , combining Lemma 1 derives that (11) holds.
Lemma 4. Assume that and are two solutions of (4) with initial data , respectively. Then, for any , it holds that where depends on , , , , , and .
Proof. We have which completes the proof.
We define as a function which is infinitely differentiable on such that , for , and . For any number , we let . Then we know that is a function in and Assume that the function is locally integrable on . We define an approximation function of as We get as almost everywhere.
We state the concept of a characteristic cone. For any , we define . Let represent the cone . We let represent the cross section of the cone by the plane , . Set , where , , and for an arbitrary . The space of all infinitely differentiable functions with compact support in is denoted by .
Lemma 5 (see [11]). Let the function be bounded and measurable in cylinder . If and , then the function satisfies .
Lemma 6 (see [11]). Let be bounded. Then the function satisfies the Lipschitz condition in and , respectively.
Using the methods presented in [11], we have the following result.
Lemma 7. If is a strong solution of problem (6), , and , it holds that where is an arbitrary constant.
Proof. Let be a twice differential function on the line . We multiply the first equation of problem (6) by the function , where . Integrating over and transferring the derivatives with respect to and to the test function , for any constant , we obtain
in which we have used .
Integration by parts yields that
Let be an approximation of the function and set . Using the properties of the , from (20) and (21), and sending , we have
which completes the proof.
3. Main Result
Generally speaking, we cannot get the boundedness of strong solutions for problem (6). This is why we assume that the strong solutions of problem (6) possess boundedness in order to establish the stability for the problem. Now we state our main result as follows.
Theorem 8. Assume that there exist strong solutions and for problem (5) or (6). Let be the maximum existence time for the solutions. If , , and the initial data , it holds that where depends on ,, , , and the coefficients of polynomial .
Proof. For , we assume that outside the cylinder
Let
where and . The function is defined in (15). Note that
Following Kruzkov’s device of doubling the variables presented in [11], from Lemma 7, and choosing , we have
Similarly, it has
It follows from (27) and (28) that
We will prove the following inequality:
We observe that the first two terms of inequality (29) can be represented in the form
From Lemma 6, we know that satisfies the Lipschitz condition in and , respectively. By the choice of , we have outside the region
Considering the estimate and the expression of function , we have
where the constant does not depend on . Using Lemma 5, we obtain as . The integral does not depend on . In fact, substituting , , , and and noting that
we have
Hence
Since
we obtain
Using Lemma 5, we have as . Using (35), we have
From (33),(37), (39), and (40), we prove that inequality (30) holds.
Set
We define
and choose two numbers and . In (30), we choose
where
When is sufficiently small, we note that function outside the cone and outside the set . For , we have the relations
Applying (41)–(45) and (30), we have the inequality
Using Lemma 4 and letting and , we obtain
By the properties of the function for , we have
where is independent of . Letting
we get
from which we obtain
Similarly, we have
It follows from (51) and (52) that
Sending and and using
from (47), (48), and (53)-(54), we have
Applying the Gronwall inequality yields the desired result.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).