PERSISTENCE PROPERTIES FOR THE GENERALIZED CAMASSA-HOLM EQUATION

. In present paper, we study the Cauchy problem for a generalized Camassa-Holm equation, which was discovered by Novikov. Our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted spaces.

1. Introduction. In this paper, we consider the following Cauchy problem of the generalized Camassa-Holm equation Recently, Novikov in [16] proposed the following integrable quasi-linear scaler evolution equation of order 2 where = 0 is a real constant. It was shown in [16] that Eq. (3) possesses a hierarchy of local higher symmetries and the first non-trivial one is Letting v(t, x) = u( t, x), then one can transform Eq. (1) into the equivalent Eq.
(1). Eq. (1) belongs to the following class [16] (1 − ∂ 2 dynamics of shallow water waves. This equation spontaneously exhibits emergence of singular solutions from smooth initial conditions. It has a bi-Hamilton structure [12] and is completely integrable [3,5]. In particular, it possesses an infinity of conservation laws and is solvable by its corresponding inverse scattering transform. After the birth of the Camassa-Holm equation, many works have been carried out to probe its dynamic properties. Such as, the Camassa-Holm equation has travelling wave solutions of the form ce −|x−ct| , called peakons, which describes an essential feature of the travelling waves of largest amplitude (see [6,8,7]). It is shown in [10,4,9] that the inverse spectral or scattering approach is a powerful tool to handle the Camassa-Holm equation and analyze its dynamics.
In [14], Yin and Li first establish the local existence and uniqueness of strong solutions for the Cauchy problem (1)- (2). Then, They prove the solution depends continuously on the initial data. Finally, they derive a blow-up criterion and present a global existence result for the equation. Recently, Mi et. al. [15] study nonuniform dependence and well-posedness for the Cauchy problem (1)-(2) in both the periodic and the nonperiodic case, respectively.
To our best knowledge, persistence properties of the Cauchy problem for (1)-(2) has not been studied yet. In this paper we first show a persistence property of the strong solutions to (1)- (2). The analysis of the solutions in weighted spaces is useful to obtain information on their spatial asymptotic behavior.
We give the definition for admissible weight function.
1. An admissible weight function for Eq. (1) is a locally absolutely continuous function φ : R → R such that, for some A > 0 and a.e. x ∈ R, |φ (x)| ≤ A|φ(x)|, and that is v-moderate for some sub-multiplicative weight function v satisfying inf R v > 0 and Our main results are stated as follows.
The restriction ab < 1 guarantees the validity of condition (5) for a multiplicative function v(x) ≥ 0. Indeed, for a < 0 one has φ(x) → 0 as |x| → ∞: the conclusion of Theorem 1.1 remains true but it is not interesting in this case, we are interested in the following two special persistence properties: (1) Power weights: Take φ = φ 0,0,c,0 with c > 0, and choose p = ∞.
Clearly, the limit case φ = φ 1,1,c,d is not covered by Theorem 1.1. In the following theorem however we may choose the weight φ = φ 1,1,c,d with c < 0, d ∈ R, and 1 |c| < p ≤ ∞, or more generally when (1 + | · |) c log(e + | · |) d ∈ L p (R). See Theorem 1.2 below, which covers the case of such fast growing weights. In other words, we want to establish a variant of Theorem 1.1 that can be applied to some v-moderate weights φ for which condition (5) does not hold. Instead of assuming (5), we now put the weaker condition where 2 ≤ p ≤ ∞.
Theorem 1.2. Let 2 ≤ p ≤ ∞ and φ be a v-moderate weight function as in Definition 1.1 satisfying condition (6) instead of (5). Let also u| t=0 = u 0 satisfy and , H s (R)), s > 3/2, be the strong solution of the Cauchy problem (1)-(2), emanating from u 0 . Then, and sup Choosing φ(x) = φ 1,1,0,0 (x) = e |x| and p = ∞ in Theorem 1.2, it follows that if |u 0 (x)| and |∂ x u 0 (x)| are both bounded by ce −|x| , then the strong solution satisfies uniformly in [0, T ]. In the following result we compute the spatial asymptotic profiles of solutions with exponential decay. As a further consequence we may infer that the peakon-like decay O(e −|x| ) mentioned above is the fastest possible decay for a nontrivial solution u of the Cauchy problem (1)-(2) to propagate for a nontrivial solution u. Theorem 1.3. Let the initial data u 0 ∈ H s (s > 3/2) and satisfy that with some d > 1/(m+1). Then the condition (8) (2)) starting from u 0 . Moreover, suppose that the functions Φ(t) and Ψ(t) satisfy with some constants c 1 , c 2 > 0 independent of t, then the following asymptotic profiles hold: The paper is organized as follows. In Section 2, we obtain a persistence result on solutions and give the proofs of Theorems 1.1-1.3.

Persistence properties of solutions.
In this section, we shall discuss the persistence properties for a generalized Camassa-Holm equation (1)-(2) in weighted L p spaces. First, using the Green function G(x) := e −|x| , x ∈ R, and the identity (1 − ∂ 2 x ) −1 f = G * f for all f ∈ L 2 , we can rewrite the Cauchy problem (1)-(2) as follows: with the operator P (D) := ∂ x (1 − ∂ 2 x ) −1 . Next, for the convenience of the readers, we present some standard definitions. In general a weight function is simply a non-negative function. A weight function v : R n → R is called sub-multiplicative if v(x + y) ≤ v(x)v(y), for all x, y ∈ R n . Given a sub-multiplicative function v, a positive function φ is v-moderate if and only if ∃C 0 > 0 : φ(x + y) ≤ C 0 v(x)φ(y), for all x, y ∈ R n . If φ is v-moderate for some sub-multiplicative function v, then we say that φ is moderate. This is the usual terminology in time-frequency analysis papers [1]. Let us recall the most standard examples of such weights. Let We have (see [2]) the following conditions: (i) For a, c, d ≥ 0 and 0 ≤ b ≤ 1 such a weight is sub-multiplicative.
The elementary properties of sub-multiplicative and moderate weights can be found in [2]. Now, we prove Theorem 1.1.
Proof of Theorem 1.1. Assume that φ is v-moderate satisfying the above conditions. Our first observation is that the first equation in (11) can be rewritten as: with the kernel G(x) = 1 2 e −|x| .
On the other hand, from the assumption u ∈ C([0, T ], H s ), s > 3/2, we get For any N ∈ Z + let us consider the N -truncations of φ(x) : f (x) = f N (x) = min{φ, N }. Then f : R → R is a locally absolutely continuous function such that Moreover, as shown in [2], the N -truncations f of a v-moderate weight φ are uniformly v-moderate with respect to N . We start considering the case 2 ≤ p < ∞. Multiply Eq. (12) by f |uf | p−2 (uf ) and integrate to obtain Note that the estimates In the first inequality we used Hölder's inequality, and in the second inequality we applied Proposition 3.1 and 3.2 in [2], and in the last we used condition (5). Here, C depends only on v and φ. Form (13) we can obtain Next, we will give estimates on u x f . Differentiating (12) with respect to x-variable, next multiplying by f produces the equation For the second order derivative term, we have where the inequality |∂ x f | ≤ Af (x), for a.e. x, is applied. Thus, it follows that Now, together the inequalities (14) with (15) and then integrating yield, At last, we treat the case p = ∞. We have u 0 , ∂ x u 0 ∈ L 2 L ∞ and f (x) = f N (x) ∈ L ∞ , hence, we have The last factor in the right-hand side is independent of q. Since f L p → f L ∞ as p → ∞ for any f ∈ L ∞ L 2 , implies that The last factor in the right-hand side is independent of N . Now taking N → ∞ implies that estimate (16) remains valid for p = ∞.
Proof of Theorem 1.2. Arguing as in the proof of Theorem 1.1, we can easily get, and  Therefore where the constants C 0 depend only on φ, b and the initial data.
Similarly, recall that ∂ x G ≤ 1 2 e −|x| and ∂ 2 Plugging the two last estimates in (17)-(18), and summing we obtain Integrating and finally letting N → ∞ yields the conclusion in the case 2 ≤ p < ∞.
The constants throughout the proof are independent of p. Therefore, for p = ∞ one can rely on the result established for finite exponents q and then let q → ∞. The rest argument is fully similar to that of Theorem 1.1.