A study on the stability of a modified Degasperis–Procesi equation

which represents a model for shallow water dynamics, has been investigated by many scholars (see Coclite & Karlsen, 2006, 2007; Degasperis, Holm, & Hone, 2002; Escher, Liu, & Yin, 2006; Lai, Yan, Chen, & Wang, 2014; Lai & Wu, 2010; Lin & Liu, 2009; Lenells, 2005; Yin, 2003). Coclite and Karlsen (2006) established existence and L stability results for entropy weak solutions of Equation (3) in the space L ⋂ BV and extended the results to a kind of generalized Degasperis–Procesi equations. Escher et al. (2006) discussed several qualitative properties of the Degasperis–Procesi equation. The existence and uniqueness of global weak solutions for Equation (1) have been established, provided that the initial data satisfy appropriate conditions in Escher (2006). Lenells (2005) dealt with the travelling wave solutions of Equation (1) and classified all weak travelling wave solutions of the Degasperis–Procesi equation. Recently, Lai et al. (2014) studied the generalized Degasperis–Procesi equation


Introduction and main results
The Degasperis-Procesi (DP) equation of the form which represents a model for shallow water dynamics, has been investigated by many scholars (see Coclite & Karlsen, 2006, 2007Degasperis, Holm, & Hone, 2002;Escher, Liu, & Yin, 2006;Lai, Yan, Chen, & Wang, 2014;Lai & Wu, 2010;Lin & Liu, 2009;Lenells, 2005;Yin, 2003). Coclite and Karlsen (2006) established existence and L 1 stability results for entropy weak solutions of Equation (3) in the space L 1 ⋂ BV and extended the results to a kind of generalized Degasperis-Procesi equations. Escher et al. (2006) discussed several qualitative properties of the Degasperis-Procesi equation. The existence and uniqueness of global weak solutions for Equation (1) have been established, provided that the initial data satisfy appropriate conditions in Escher (2006). Lenells (2005) dealt with the travelling wave solutions of Equation (1) and classified all weak travelling wave solutions of the Degasperis-Procesi equation. Recently, Lai et al. (2014) studied the generalized Degasperis-Procesi equation where k ≥ 0 and m > 0 are constants. Lai et al. (2014) derived the L 2 (R) conservation law and established the L 1 (R) stability of local strong solutions to Equation (2) by assuming that its initial

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In this paper, we have studied the generalized Degasperis-Procesi equation. Using the partial differential operator, the equivalent form Equation (6) has been derived. In the previous sections, we have used the Kato Theorem and double variables method to establish existence and stability of the solution for Equation (6) . The approaches presented in this paper can be summarized to discuss other partial differential equations with initial value. . For other approaches to study the DP equation and related partial differential equations, the reader is referred to Kato (1975), Kruzkov (1970), Rodriguez-Blanco (2001) and the references therein.
The objective of this work is to investigate the modified Degasperis-Procesi equation in the form where , ∈ R, m ≥ 2, function f ( ) is a polynomial of order n(n ≥ 2). Letting = 0, = 4, f ( ) = Assuming that the initial value (0, x) of Equation (4) belongs to H s (R) s > 3 2 , we will prove the existence and uniqueness of local solution for Equation (4) using the Kato Theorem (see Kato, 1975) Furthermore, we will use the approaches presented in Kruzkov (1970) to establish the L 1 (R) local stability of the solution for the modified Degasperis-Procesi Equation (4). The results obtained in this paper extend parts of results presented in Lai et al. (2014).
We let H s (R) (where s is a real number) denote the Sobolev space with the norm defined by For simplicity, we let c denote any positive constants.
We consider the Cauchy problem of Equation (3) which is equivalent to the problem Now we state the main results of our work.
Theorem 1.2 Assume that (t, x) and (t, x) are two local strong solutions of problem (5) or (6) with initial data 0 (x), 0 (x) ∈ L 1 (R) ∩ H s (R)(s > 3 2 ), respectively. Let T > 0 be the maximum existence time of (t, x) and (t, x). For any t ∈ [0, T), it holds that This paper is organized as follows. Section 2 gives the proof of Theorem 1.1. The proof of Theorem 1.2 is given in Section 3.

Proof of Theorem 1.1
Firstly, we introduce the abstract quasi-linear evolution equation Let X and Y be Hilbert spaces where Y is continuously and densely embedded in X, and Q:Y → X be a topological isomorphism. We define L(Y, X) as the space of all bounded linear operators from Y to X.
We denote L(X, X) by L(X). Note that 1 , 2 , 3 and 4 in the following are constants and depend on max{‖y‖ Y , ‖z‖ Y }.
(II) QA(y)Q −1 = A(y) + B(y), in which B(y) ∈ L(X) is bounded and uniform on bounded sets in Y and (III) W:Y → Y extends to a map from X to X, is bounded on bounded sets in Y and satisfies Kato Theorem (1975). Assume that (I), (II) and (III) hold. If u 0 ∈ Y, there is a maximal T > 0 depending only on ‖u 0 ‖ Y and a unique solution u to problem (7) such that x f ( ), and Q = Λ. Then, we will verify that A( ) and W( ) satisfy conditions (I)-(III). We cite several conclusions presented in Rodriguez-Blanco (2001).
Lemma 2.4 (Kato, 1975). Let r and q be real numbers such that −r < q ≤ r. Then,

. Then, W is bounded on bounded sets in H s and satisfies
Proof For s > 3 2 , we have ‖ ‖ L ∞ ≤ c‖ ‖ H s and ‖ ‖ H s−1 ≤ c‖ ‖ H s. Applying the algebra property of H s (R) and Lemma 2.4, we get which completes the proof of (8). Similarly, we get which completes the proof of (9). ✷ Proof of Theroem 1.1 Using Lemmas 2.1-2.3 and Lemma 2.5, we know that the conditions (I), (II) and (III) hold. Applying the Kato Theorem, we find that problem (5) or (6) where positive constant c depends on , , m, n, ‖ 0 ‖ L ∞ and k = max{m, n}.
Proof Using the property of the operator Λ −2 and Remark 2.6, we get in which we apply the Tonelli Theorem to complete the proof. ✷ We introduce a function ( ) which is infinitely differential on (−∞, +∞). Note that ( ) ≥ 0, We Lemma 3.3 (Kruzkov, 1970). Let the function v(t, x) be bounded and measurable in cylinder [0, T 1 ] × K r . If for any ∈ (0, min[r, T 1 ]) and any ∈ (0, ), the function satisfies Lemma 3.4 (Kruzkov, 1970). If F(u) u is bounded, then the function H(u, v) = sign(u − v)(F(u) − F(v)) satisfies the Lipschitz condition in u and v.
We state the concept of a characteristic cone. Let T be described in Theorem 1.1 and ‖ ‖ L ∞ (R) ≤ M T . For any T 1 ∈ [0, T) and R 1 > 0, we define Let Ω represent the cone (t, x):|x| ≤ R 1 − Nt, 0 ≤ t ≤ T 0 = min(T 1 , R 1 N −1 ) and S designate the cross section of the cone Ω by the plane t = , ∈ [0, T 0 ].
where k is an arbitrary constant.