The "Continual" Study Of Two Degrees Of Degasperis - Procesi Equations In Weighted Space

In this article, we study the Cauchy problem of the 2-component Degasperis-Procesi (2-DP) equations This system was firstly proposed by Popowicz as the Hamiltonian extension of the DP equation in [38]. In Eq.(1) u(x, t) describes the horizontal velocity of the flu-id while p(x, t) describes the horizontal deviation of the surface from equilibrium. u and p are both measured under dimensionless units. The persistence proper-ty for solutions of Eq.(1) in exponential weighted space was investigated in [19]. Motivated by [1,2], we obtain more general results about persistence property by extending the notion of weight function. This paper is organized as follows. In Chapter 1 we introduce some background knowledge on Eq.(1) and some history in shallow water wave regime. In Chapter 2 we present some preliminaries, such as weight functions and relevant weighted Young inequality. Using the method of characteristics, persistence results for solutions of Eq.(1) in weighted LP space and asymptotic profiles are obtained in Chapter 3. The main results read as follows.Theorem 1.2 Let s≥ 2,2≤ p≤∞,0< T< T*, and is the unique strong solutions of (1), such that the initial value u0(x)= u(x,0), ρo(x) =ρ(x,0) satisfies then, we have the estimate where Φ is the admissible weight function in Definition 2.2.Theorem 1.3 Let s> 2,2≤ p≤ oo and (u, ρ) is the unique strong solutions of (1),Φ is the admissible weight function in Definition 2.2 satisfying condition (2.2) instead of (1.6). If the initial data of (1) satisfies thenWhen choosing a proper weight function and using the above conclusions, we can obtain the following asymptotic profiles:Corollary 1.4 Assuming s> 5/2, the initial value (u0, ρ0) ∈ (R)×Hs-1(R), and (u0,ρ0)(?) 0. If the weight function satisfies then the strong solutions (u, ρ) of (1) satisfies and the above formula is uniform in t ∈ [0, T]. Moreover, we have where Φ(t),ψ(t) are continuous in [0,T].