The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces

This paper deals with the Cauchy problem for a generalized $b$-equation with higher-order nonlinearities 
 $y_{t}+u^{m+1}y_{x}+bu^{m}u_{x}y=0$, where $b$ is a constant and $m\in\mathbb{N}$, the notation $y:= (1-\partial_x^2) u$, which includes the famous $b$-equation and Novikov equation as special cases. The local well-posedness in critical Besov space $B^{3/2}_{2,1}$ is established. Moreover, a lower bound for 
the maximal existence time is derived. Finally, the persistence properties in weighted $L^p$ spaces for the solution of this equation are considered, which extend the work of Brandolese [L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not. 22 (2012), 5161-5181] on persistence properties to more general equation with higher-order nonlinearities.

1. Introduction. The present paper focuses on the Cauchy problem for the following shallow water equation with high-order nonlinearities y t + u m+1 y x + bu m u x y = 0, x ∈ R, t > 0, u(x, 0) = u 0 (x), x ∈ R, (1.1) where b is a constant and m ∈ N, the notation y := (1 − ∂ 2 x )u. It is easy to see that model (1.1) contains the two kinds of famous shallow water equation, that is, b-equation and Novikov equation. Our main purpose of this paper is to establish the well-posedness in critical Besov space B 3/2 2,1 and persistence in weighted Sobolev space.
Obviously, if m = 0, b ∈ R, the Equ.(1.1) becomes a b-equation, which can be derived as the family of asymptotically equivalent shallow water wave equations that emerges at quadratic order accuracy for any b = −1 by an appropriate Kodama transformation. For the case b = −1, the corresponding Kodama transformation is singular and the asymptotic ordering is violated (see [29,30,31]).

SHOUMING ZHOU
Equ. (1.2) belongs to the following family of nonlinear dispersive partial differential equations u t − γu xxx − α 2 u xxt = (c 1 u 2 + c 2 u 2 x + c 3 uu xx ) x , where γ, α, c 1 , c 2 and c 3 are real constants. By using Painlevé analysis in [26,28,45], there are only three asymptotically integrable within this family: the KdV equation, the Camassa [41,42], where b was taken as a bifurcation parameter. The necessary conditions for integrability of the b-equation were investigated in [55]. The b-equation also admits peakon solutions for any b ∈ R (see [28,41,42]). The well-posedness, blow-up phenomena and global solutions for the b-equation were shown in [34,37,58].
In fact, the Camassa-Holm and Degasperis-Procesi equations are the only integrable members of the b-equation family with a bi-Hamiltonian structure [45]. The Camassa-Holm equation arises in a variety of different contexts. In 1981, it was originally derived as a bi-Hamiltonian equation with infinitely many conservation laws by Fokas and Fuchssteiner [35]. It has been widely studied since 1993 when Camassa and Holm [6] proposed it as a model for the unidirectional propagation of shallow water waves over a flat bed. Such as, Camassa-Holm equation has also a bi-Hamiltonian structure [35,50] and is completely integrable [6,11,46], and it possesses an infinity of conservation laws and is solvable by its corresponding inverse scattering transform (cf. [5,9,16]). The stability of the smooth solitons was considered in [22], and the orbital stability of the peaked solitons were proved in [21], It is worth pointing out that solutions of this type are not mere abstractions: the peakons replicate a feature that is characteristic for the waves of great heightwaves of largest amplitude that are exact solutions of the governing equations for irrotational water waves (see [15] and references therein). The explicit interaction of the peaked solitons were given in [2]. It has been shown that this problem is locally well-posed for initial data u 0 ∈ H s with s > 3 2 (cf. [12,51,23]). Moreover, Camassa-Holm equation not only has global strong solutions, but also admits finite time blow-up solutions [10,12,13,51,17], and the blow-up occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. On the other hand, it also has global weak solutions in H 1 (see [3,14,20,64]). The advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solitons and models the peculiar wave breaking phenomena (cf. [7,13]).
In 1999, Degasperis and Procesi [26] derived a nonlinear dispersive equation, which can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as that for the Camassa-Holm shallow water equation, and the Degasperis-Procesi equation can be obtained from the shallow water elevation equation by an appropriate Kodama transformation(cf. [19,29,30]). Degasperis, Holm and Hone [27] proved the formal integrability of this equation by constructing a Lax pair. They also showed that this equation has a bi-Hamiltonian structure and an infinite sequence of conserved quantities, and admits exact peakon solutions. Lundmark and Szmigielski [54] presented an inverse scattering approach for computing n-peakon solutions, and the direct and inverse scattering approach pursued recently in [18]. The traveling wave solutions for the Degasperis-Procesi equation was studied by Vakhnenko and Parkes in [61]. Similar to the Camassa-Holm equation, the local well-posedness, the precise blow-up scenario and the global existence of strong solutions to Degasperis-Procesi were derived in [32,53,66,67]. On the other hand, it also has global weak solutions in H 1 (see [32,66]) and global entropy weak solutions belonging to the class L 2 ∩ BV and to the class L 2 ∩ L 4 (see [8]). A special property of the Degasperis-Procesi equation is the existence of discontinuous shock wave and periodic shock wave solutions (see [33] and references therein).
On the other hand, taking m = 1, b = 3 in (1.1) we arrived the Novikov equation, which was recently discovered by Novikov in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity [59]. The perturbative symmetry approach [55] yields necessary conditions for a PDE to admit infinitely many symmetries. Using this approach, Novikov was able to isolate the Equ.(1.3) and find its first few symmetries, and he subsequently found a scalar Lax pair for it, proving that the equation is integrable. By using the prolongation algebra method, Hone and Wang [43] gave a matrix Lax pair and many conserved densities and a bi-Hamiltonian structure of the Novikov equation, and showed how it was related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. The explicit formulas for multipeakon solutions of Novikov equation were derived in [44]. Recently, by the transport equations theory and the classical Friedrichs regularization method, the authors proved that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces B s p,r (with 1 ≤ p, r ≤ +∞ and s > max{1+1/p, 3/2} in [62,65], and with the critical index s = 3/2, p = 2, r = 1 in [56]). It is also shown in [56] that the Novikov equation associated with the initial value is locally well-posed in Sobolev space H s with s > 3/2 by using the abstract Kato Theorem. Two results about the persistence properties of the strong solution for Equ.(1.3) were established in [56]. A Galerkin-type approximation method was used to establish the well-posedness of Novikov equation in the Sobolev space H s with s > 3/2 on both the line and the circle [39], and in [48,68] the authors proved that the data-to-solution map is not globally uniformly continuous on H s for s < 3/2, this result supplements Himonas and Holliman's works [39]. Tiglay [60] shown the local well-posedness of the problem in Sobolev spaces and existence and uniqueness of solutions for all time using orbit invariants. For analytic initial data, the existence and uniqueness of analytic solutions for Equ. (1.3) were also obtained in [60]. Analogous to the Camassa-Holm equation, the Novikov equation possesses blow-up phenomenon [47,68] and global weak solutions [49,63].
Motivated by the results mentioned above, in [69,70], we considered the Cauchy problem for a weakly dissipative shallow water equation with higher-order nonlinearities where λ, b are constants and m ∈ N, the notation y := (1 − ∂ 2 x )u. The local well-posedness of solutions for the Cauchy problem in Besov space B s p,r with 1 ≤ p, r ≤ +∞ and s > max{1 + 1 p , 3 2 } is obtained in [70], and the global existence and blow-up phenomenon propagation behaviors of compactly supported solutions are also established in [70]. The persistence properties of the strong solutions, the existence of weak solutions and the explicit formulas for peakon and multipeakon for Equ.(1.4) with λ = 0 are studied in [69]. The local and global existence and 2,1 ) is continuous. [24]) , Theorem 1.1 holds true in the case of B 1+1/p p,1 with 1 ≤ p < ∞. Besides, using the similar arguments in [23], Theorem 1.1 can also hold true in the case of B s p,r with s > max(1 + 1/p, 3/2). Furthermore, the existence of solutions to Equ.(1.1) holds as the initial data belong to B s p,r ∩ Lip with s > 1, which improves the corresponding result in [70].  Our main blow-up criterion reads: Let u 0 ∈ Lip ∩ B s p,r with 1 ≤ p, r ≤ ∞ and s > max(3/2, 1 + 1/p), then there exists a lifespan T * u0 > 0 such that Moreover, if b = m + 2 and u 0 ∈ H 1 , then Next, we get a lower bound for the maximal existence time which depending only on ||u 0 || Lip .
p,r , s > max{3/2, 1 + 1/p}. Let T * be the maximal existence time of the solution u to Equation (1.1) with the initial data u 0 . Then T * satisfies where a depending only on m and |b|.
In [4,38,40,56,57,69], the spacial decay rate for the strong solution to the Camassa-Holm [4,40,57], b-equation [38], Novikov equation [56] were established provided that the corresponding initial datum decays at infinity. This kind of property is so-called persistence property. In the present paper, we intend to find a large class of weight functions φ such that where || · || p denotes the usual L p norm. this way we obtain a persistence result on solutions u to Equ.(1.1) in the weight L p spaces L p , φ := L p (R, φ p dx). As a consequence and an application we determine the spatial asymptotic behavior of certain solutions to Equ.(1.1). Our results generalize the work of [4] on persistence and non-persistence of solutions to (1.1) in L p,φ . We will work with moderate weight functions which appear with regularity in the theory of time-frequency analysis [1,36] and have led to optimal results for the Camassa-Holm equation in [4], we first give the definition for admissible weight function Definition 1.1. An admissible weight function for the Equ.(1.1) 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 func- (1.6) We can now state our main result on admissible weights.
be a strong solution of the Cauchy problem for Equ.
, where φ is an admissible weight function for the Equ.(1.1). Then, for all t ∈ [0, T ], we have the estimate The basic example of the application of Theorem 1.5 is obtained by taking the standard weights φ = φ a,b,c,d (x) = e a|x| b (1 + |x|) c log(e + |x|) d with the following conditions: a ≥ 0, c, d ∈ R, 0 ≤ b ≤ 1, ab < 1. The restriction ab < 1 guarantees the validity of condition (1.6) for a multiplicative function v(x) ≥ 1. Indeed, for a < 0 one has φ(x) → 0 as |x| → ∞: the conclusion of the Theorem 1.5 remains true but it is not interesting in this case. We interesting the following two special cases: (1) Take φ = φ 0,0,c,0 with c > 0, and choose p = ∞. In this case the Theorem 1.6 states that the condition Thus, Theorem 1.5 generalizes the main result of Ni and Zhou [57] on algebraic decay rates of strong solutions to the Equ.(1.1). ( It is easy to see that such weight satisfies the admissibility conditions of Definition 1. Clearly, the limit case φ = φ 1,1,c,d is not covered by Theorem 1.5. 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 1.6 below, which covers the case of such fast growing weights. In other words, we want to establish a variant of Theorem 1.5 that can be applied to some v-moderate weights φ for which condition (1.6) does not hold. Instead of assuming (1.6), we now put the weaker condition Theorem 1.6. Let 2 ≤ p ≤ ∞ and φ be a v-moderate weight function as in Definition 1.1 satisfying condition (1.7) instead of (1.6). Let also u| t=0 = u 0 satisfy , H s (R)), s > 3/2 be the strong solution of the Cauchy problem for Equation (1.1), emanating from u 0 Then, and sup

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Choosing φ(x) = φ 1,1,0,0 (x) = e |x| and p = ∞ in Theorem 1.6. It follows that if |u 0 (x)| and |∂ x u 0 (x)| are both bounded by ce −|x| , then the strong solution satisfies 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 Equ.(1.1) to propagate.
with some constants c 1 , c 2 > 0 independent on t, then the following asymptotic profiles hold: for all t ∈ [0, T ] provided that either (i) m = 0 and 0 < b < 3, or (ii) b = m + 1 and m is a even.
The plan of this paper is organized as follows. In the next section, the local wellposedness in critical Besov space B By the arguments similar to the case u 0 ∈ B s p,r , s > max{1 + 1 p , 3 2 }(see [24,70]), we can easy get the following two lemmas: initial data u 0 (respectively v 0 ). Then for every t ∈ [0, T ]: Hence Φ is Hölder continuous from B  Recall that v (n) .
= ∂ x u (n) solves the linear transport equation: Thanks to the Kato theory [23], . According to the first step, we have that the sequence u 2,1 ), thus we can use Proposition 3 in [24], which implies that w (n) tends to w ∞ in C([0, T ]; B On the other hand, applying Lemma 2.3 in [56] and the product law in the Besov spaces to equation (2.5), one may get that dτ . (2.6) Using the properties of Besov spaces exhibited in [24], one easily checks that (f (n) ) n∈N is uniformly bounded in C([0, T ]; B 1 2 1,2 ). Moreover, Hence, combining the convergence of z (n) in C([0, T ]; B 1 2 2,1 ) with estimates (2.4)-(2.7), we deduce that for large enough n ∈ N dτ .

SHOUMING ZHOU
Thanks to the Gronwall's inequality, we have
Proof. Applying the last of Lemma 2.4 in [70] to Equ.(1.1) and using the fact that As s − 1 > 0, according to Lemma 2.5 in [70], one gets Applying Gronwall lemma completes the proof of (3.1). By differentiating once equation (1.1) with respect to x, and applying the L ∞ estimate for transport equations, we easily prove that x ) −1 f = 1 2 e −|x| * f and the Young inequality, we get Lip for some universal constant C . Hence Gronwall lemma gives inequality (3.2).
Proof of Theorem 1.3. Applying Theorem 1.1, there exists a unique solution u to Equ.(1.1) with the initial data u 0 . Assume that u satisfies Hence, for all t ∈ [0, T * ), (3.1) insures that Let > 0 be such that 2(m + 1)C M m T * < 1 where C stands for the constants used in the proof of Proposition 3.1 in [70]. We then have a solutionũ(t) ∈ E s p,r ( ) to Equ.(1.1) with initial datum u(T * − /2). For the sake of uniqueness,ũ(t) = u(t + T * − /2) on [0, /2) so thatũ extends the solution u beyond T * . We conclude that T * < T * u0 . Multiply Equ.(1.1) by u, we have Integrating by parts on R, Clearly, if b = m + 2, we get ||u|| 2 Similar to the above argument, we have This completes the proof of Theorem 1.3.
Proof of Theorem 1.4. Multiplying Equation (2.1) by u 2n−1 with n ∈ Z + and integrating by parts, we obtain Note that the estimates are true. Moreover, using Hölder's inequality Thus, we can obtain Since ||f || L n → ||f || L ∞ as n → ∞ for any f ∈ L ∞ ∩L 2 and the operator (1−∂ 2 Form above inequality we deduce that Next, we will give estimates on ||u x (x, t)|| L ∞ . Differentiating (2.1) with respect to x-variable produces the equation Similar to the estimate of (3.4), we deduce that where a depending only on m and |b|. Define T := 1 a(2||u0|| L ∞ +||u0,x|| L ∞ ) m+1 . By (3.6), then for all t < min{T, T * }, one can easily get Given a sub-multiplicative function v, a positive function φ is v-moderate if and only if 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 (1 + |x|) c log(e + |x|) d .
We have (see [4]) the following conditions: (i) For a, c, d ≥ 0 and 0 ≤ b ≤ 1 such weight is sub-multiplicative.
The elementary properties of sub-multiplicative and moderate weights can be find in [4]. Next, we prove Theorem 1.5.
Proof of Theorem 1.5. We define We also introduce the kernel G(x) = 1 2 e −|x| . Then the Equ.(1.1) can be rewritten as u t + u m+1 ∂ x u + G * E(u) = 0, (4.1) Note that, from the assumption u ∈ C([0, T ], H s ), s > 3/2, we get For any N ∈ Z + let us consider the N -truncations Observe that f : R → R is a locally absolutely continuous function such that In addition, if C 1 = max{C 0 , α −1 }, where α = inf x∈R v(x) > 0, then Indeed, let us introduce the set . The constant C 1 being independent on N , this shows that the N -truncations of a v-moderate weight are uniformly v-moderate with respect to N .
We start considering the case 2 ≤ p < ∞. Multiplying the Equation (4.1) by f , and then by |uf | p−2 (uf ) we get, after integration, In the first inequality we used Hölder's inequality, and in the second inequality we applied Proposition 3.1 and 3.2 in [4], and the last we used condition (1.6). Here, C depends only on v and φ. Form (4.2) we can obtain Next, we will give estimates on u x f . Differentiating (4.1) with respect to xvariable, next multiplying by f produces the equation Multiplying this equation by |f ∂ x u| p−2 (f ∂ x u) with p ∈ Z + , integrating the result in the x-variable, and note that In the third inequality we applied the pointwise bound |∂ x G(x)| ≤ 1 2 e −|x| and the condition.
In the last inequality we used |∂ x f (x)| ≤ Af (x) for a.e. x. Thus, we get Now, combing the inequalities (4.3) with (4.4) and then integrating yields, for all t ∈ [0, T ]. Since f (x) = f N (x) ↑ φ(x) as N → ∞ for a.e. x ∈ R. Recall that u 0 φ ∈ L p (R) and ∂ x u 0 φ ∈ L p (R), we get 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 (4.5) The last factor in the right-hand side is independent on 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 on N . Now taking N → ∞ implies that estimate (4.5) remains valid for p = ∞.
Plugging the two last estimates in (4.6)-(4.7), and summing we obtain Integrating and finally letting N → ∞ yields the conclusion in the case 2 ≤ p < ∞. The constants throughout the proof are independent on 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 1.5.

SHOUMING ZHOU
Next, We show that the last term in (4.8) can be included inside the lower order terms of the asymptotic profiles (1.11). The assumption m = 0 or b = m+ 1 ensures the validity of the following equality: By the arguments in [4], this achieves the asymptotic representation of u.