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

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


Introduction
The Degasperis-Procesi (DP) equation V  − V  + 4VV  = 3V  V  + VV  ,  > 0,  ∈ , (1) 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 , ,  0 ,  1 ,  2 ,  3 ∈ .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 Kortewegde 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][2][3][4][5][6][7][8][9][10][11]).For example, Coclite and Karlsen [3] established the well-posedness of  1 ∩ 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  > 3/2.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 (, ) ∈  0 (), (, ) ∈  3 0 (), and function (⋅) is a polynomial of order  ( ≥ 2).Letting  = 4,  = 1, (V) = V 2 /2, (5) reduces to the Degasperis-Procesi equation (1).We consider the Cauchy problem of ( 5) with an initial condition V 0 ().Namely, Assume that (5) possesses a bounded strong solution in its maximum existence time interval [0, ) and V 0 lies in  1 () ∩   () ( > 3/2).We use the approaches of Kružkov doubling the variables presented in [12] to prove the  1 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.
Proof.We have in which we have used ∫   −|−|  = 2 to complete the proof.
Proof.Using the property of the operator Λ −2 , we get in which we apply the Tonelli Theorem to complete the proof.
Proof.Suppose that Φ(V) is a twice differential function.
Multiplying the first equation of problem ( 8) by Φ  (V)(, ) and integrating over   , we get Using the method of integration by parts, we get Notice that Then we have Mathematical Problems in Engineering Substituting ( 23) and ( 26) into (22), we get Let Φ  (V) be an approximation of the function combining with (27), we complete the proof.