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A few remarks on the generalized Novikov equation
Journal of Inequalities and Applications volume 2013, Article number: 560 (2013)
This paper deals with the Cauchy problem for a generalized Novikov equation , where b is a constant. The local well-posedness in the critical Besov space is established. Moreover, a lower bound for the maximal existence time and lower semicontinuity of the existence are derived, the multi-peakon solutions are also obtained. Finally, the persistence properties in weighted spaces for the solution of this equation are considered.
MSC:35G25, 35L05, 35Q50.
The present paper focuses on the Cauchy problem for the following modified Novikov equation:
where b, and are arbitrary constants. Our main purpose of this paper is to establish the well-posedness in the critical Besov space and persistence in a weighted Sobolev space.
Note that when we take , Eq. (1.1) is the Novikov equation:
which was recently discovered by Novikov in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity . The perturbative symmetry approach  yields necessary conditions for a PDE to admit infinitely many symmetries. Using this approach, Novikov was able to isolate Eq. (1.2) 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  gave a matrix Lax pair and many conserved densities and a bi-Hamiltonian structure of the Novikov equation, and they showed how it was related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. Then in , the authors calculated the explicit formulas for multi-peakon solutions of the Novikov equation.
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 (with and in [5, 6], and with the critical index , in ). It was also shown in  that the Novikov equation associated with the initial value is locally well posed in the Sobolev space with by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for Eq. (1.2) were established in . A Galerkin-type approximation method was used in Himonas and Holliman’s paper  to establish the well-posedness of Novikov equation (1.2) in the Sobolev space with on both the line and the circle, and in [9, 10] the authors proved that the data-to-solution map is not globally uniformly continuous on for , this result supplements Himonas and Holliman’s works. Tiglay  showed the local well-posedness of the problem in Sobolev spaces and the existence and uniqueness of solutions for all time using orbit invariants. For analytic initial data, the existence and uniqueness of analytic solutions for Eq. (1.2) were also obtained in . Analogous to the Camassa-Holm equation, the Novikov equation possesses a blow-up phenomenon [10, 12] and global weak solutions [13, 14].
On the other hand, it is well known that the nonlinearity of the following b-equation is quadratic:
which can be derived as the family of asymptotically equivalent shallow water wave equations that emerges at quadratic order accuracy for any by an appropriate Kodama transformation. For the case , the corresponding Kodama transformation is singular and the asymptotic ordering is violated (see [15–17]). Equation (1.2) belongs to the following family of nonlinear dispersive partial differential equations:
where γ, α, , and are real constants. By using Painlevé analysis in [18–20], there are only three asymptotically integrable within this family: the KdV equation, the Camassa-Holm (Eq. (1.3) with ) equation and the Degasperis-Procesi equation (Eq. (1.3) with ). The solutions of the b-equation were studied numerically for various values of b in [21, 22], where b was taken as a bifurcation parameter. The necessary conditions for integrability of the b-equation were investigated in . The b-equation also admits peakon solutions for any (see [19, 21, 22]). The well-posedness, blow-up phenomena and global solutions for the b-equation were shown in [23–25].
Recently, Mi and Mu  studied the local well-posedness in the Besov space with and . It is well known that and for any : , which shows that and are quite close, so here we first establish the local well-posedness in the critical Besov space .
Theorem 1.1 Assume that the initial data . Then there exist a unique solution and a maximal time to the Cauchy problem (1.1) such that
Moreover, the solution depends continuously on the initial data, i.e., the mapping
Remark 1.1 Following the proof of Theorem 5.1 in  and Theorem 1.3 in , one can easily get that Eq. (1.1) is not locally well posed in in the following sense: There exists a global solution to Eq. (1.1) such that for any and , there exists a solution to Eq. (1.1) with
Remark 1.2 Theorem 1.1 improves the corresponding result in . On the other hand, noting that the counterexample given in  cannot be applied to the case in with , the question of local well-posedness of Eq. (1.1) in remains open. Actually, this is still an open problem for the Camassa-Holm and Novikov equations.
Remark 1.3 Since
holds for (see ), Theorem 1.1 holds true in the case of with . Besides, using similar arguments in , Theorem 1.1 can also hold true in the case of with . Furthermore, the existence of solutions to Eq. (1.1) holds as the initial data belong to with , which covers the corresponding result in .
Theorem 1.2 Let with and , then there exists a lifespan such that
We now get a lower bound depending only on for the maximal existence time.
Theorem 1.3 Assume that , . Let be the maximal existence time of the solution u to Eq. (1.1) with the initial data . Then satisfies
Next, we shall derive lower semicontinuity of the existence time, provided the initial data is smooth enough.
Theorem 1.4 Let , and . Assume that u, v are two solutions to Eq. (1.1) with the initial data , . Let , be the maximal existence time of the solution u, v. If there exist and a constant C such that
then Eq. (1.1) has a unique solution .
In , the authors consider the single peakon taking the form , . Moreover, this peakon solitary is a global weak solution to Eq. (1.1). Next, we discuss the existence of multi-peakon solutions to Eq. (1.1).
Theorem 1.5 Equation (1.1) has peakon solutions of the form:
whose positions and amplitudes are according to the dynamical system
In [7, 29–33], the spacial decay rates for the strong solutions to the Camassa-Holm equation, the b-equation, and the Novikov equation were established provided that the corresponding initial data decay at infinity. This kind of property is the so-called persistence property. Following the main idea of , we also prove the persistence properties in weighted spaces for the solution of Eq. (1.1). However, the hard question is that there are cubic nonlinearities in (1.1) which make the proof very difficult. First, we give the following definition of an admissible weight function.
Definition 1.1 An admissible weight function for Eq. (1.1) is a locally absolutely continuous function such that, for some and a.e. , , and that is v-moderate for some sub-multiplicative weight function v satisfying and
We recall that a locally absolutely continuous function is a.e. differentiable in ℝ. Moreover, its a.e. derivative belongs to and agrees with its distributional derivative. We can now state our main result on admissible weights.
Theorem 1.6 Let , , and . Let also be a strong solution of the Cauchy problem for Eq. (1.1) such that satisfies
where ϕ is an admissible weight function for Eq. (1.1). Then, for all , we have the estimate
for some constant depending only on v, ϕ (through the constants A, , , and ), and
Remark 1.4 The basic example of the application of Theorem 1.6 is obtained by taking the standard weights with the following conditions:
(For , one has as : the conclusion of the theorem remains true but it is not interesting in this case.) The restriction guarantees the validity of condition (1.6) for a multiplicative function .
The limit case is not covered by Theorem 1.1. The result holds true, however, for the weight with , , and , or, more generally, when . See Theorem 1.6 below, which covers the case of such fast growing weights.
Remark 1.5 Let us consider a few particular cases:
Take with , and choose . In this case, Theorem 1.6 states that the condition
implies the uniform algebraic decay in :
It is worth pointing out that this is a new result for the Novikov equation.
Choose if and if with . Such weight clearly satisfies the admissibility conditions of Definition 1.1. Applying Theorem 1.6 with , we conclude that the pointwise decay as is conserved during the evolution. Similarly, we have persistence of the decay as . Hence, our Theorem 1.6 encompasses also Theorem 4.1 of .
Since ‘peakon’ solution , does not satisfy the asymptotic behavior in Theorem 1.2 (see Remark 1.4), the purpose of the next theorem is to establish a variant of this theorem 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
It is easily checked that, for any continuous sub-multiplicative weight function v, we have
so that condition (1.7) is indeed weaker than condition (1.6) (see  for the details).
Theorem 1.7 Let and ϕ be a v-moderate weight function as in Definition 1.1 satisfying condition (1.7) instead of (1.6). Let also satisfy
Let also , be the strong solution of the Cauchy problem for Eq. (1.1), emanating from . Then
The plan of this paper is organized as follows. In the next section, the local well-posedness in the critical Besov space is considered and Theorem 1.1 is proved. The blow-up criteria and multi-peakon solutions are obtained in Section 3 and Theorems 1.2-1.5 are proved. In the last section, the persistence properties in weighted spaces for the solution of Eq. (1.1) are considered, and Theorems 1.6-1.7 are proved.
2 Local well-posedness in critical Besov spaces
In this section, we shall establish the local well-posedness of Eq. (1.1) in critical Besov spaces. More precisely, we give the proof of Theorem 1.1. First, we rewrite model (1.1) in the following transports equation form:
We can easily get the following two lemmas.
Lemma 2.1 Let . Then there exists a time such that the Cauchy problem (1.1) has a solution .
Lemma 2.2 Assume that such that is a solution to the Cauchy problem (1.1) with the initial data (respectively ). Let and . Then, for every ,
Lemma 2.3 For any , there exist a neighborhood V of in and a time such that for any solution of the Cauchy problem (1.1) , the map
Proof Firstly, we prove the continuity of the map Φ in . Fix and . Now we claim that there exist and such that for any with , the solution of the Cauchy problem (1.1) belongs to and satisfies . In fact, according to the proof of the local well-posedness, we have that if we fix such that
As , then . Here, one can choose some suitable constant C such that
Now, combining the above uniform bounds with Lemma 2.2, we get that
Hence Φ is Holder continuous from into .
Next we prove the continuity of the map Φ in . Let and in . Let be the solution of the Cauchy problem (1.1) with the initial data . From the above argument, we deduce that for any , ,
Note that to prove in means to prove in .
Recall that solves the linear transport equation:
Thanks to the Kato theory , we decompose into with
According to the first step, we have that the sequence () is uniformly bounded in and tends to in , thus we can use Proposition 3 in , which implies that tends to in , i.e., for any , .
Using the properties of Besov spaces exhibited in , one easily checks that is uniformly bounded in . Moreover,
Hence, combining the convergence of in with estimates (2.4)-(2.7), we deduce that for large enough ,
Thanks to Gronwall’s inequality, we have
for some constant C depending only on M and b. We have completed the continuity of the map Φ in .
Now, applying to Eq. (1.1) and by the same argument to the resulting equation in terms of , we may check the continuity of the map Φ in . □
Proof of Theorem 1.1 Combining the result in Lemma 2.1 with that in Lemma 2.2, one gets the existence and uniqueness of the solution of the Cauchy problem (1.1). And Lemma 2.3 shows that the solution of the Cauchy problem (1.1) depends continuously on the initial data. This completes the proof of Theorem 1.1. □
3 Blow-up criterion and multi-peakon solutions
This section is devoted to the proof of Theorems 1.2-1.5. Theorems 1.2-1.3 are based on the following lemma.
Lemma 3.1 Let with and . Let solve Eq. (1.1) on with the initial data . There exist a constant depending only on s and p and a universal constant such that for all , we have
Proof Applying the last of Lemma 2.3 in  to the Novikov equation and using the fact that is a multiplier of order −2 yields
As , according to Lemma 2.2(5) in , one gets
Applying Gronwall’s lemma completes the proof of (3.1).
By differentiating once Eq. (1.1) with respect to x, and applying the estimate for transport equations, we easily prove that
Since and the Young inequality, we get
for some universal constant . Hence Gronwall’s lemma gives inequality (3.2). □
Proof of Theorem 1.2 Let be such that is finite. According to inequality (3.2), is also finite. Hence, (3.1) insures that
Let be such that , where C stands for the constants used in the proof of Lemma 2.1 in . We then have a solution to Eq. (1.1) with the initial data . For the sake of uniqueness, on so that extends the solution u beyond . We conclude that and Theorem 1.2 is proved. □
Proof of Theorem 1.3 Multiplying Eq. (2.1) by with and integrating by parts, we obtain
with . Note that the estimates
are true. Moreover, using Hölder’s inequality
from (2.2) we can obtain
Since as for any and the operator , from the above inequality we deduce that
Next, we give estimates on . Differentiating (2.1) with respect to x-variable produces the equation
Similar to the estimate of (3.4), we deduce that
Combining (3.4) with (3.5), we get
with . Define . By (3.6), then for all , one can easily get
Theorem 1.2 yields that . This completes the proof of Theorem 1.3. □
Proof of Theorem 1.4 Let . In view of Eq. (2.1), one can get
Using standard energy arguments and integration by parts, we end up with
Using the above inequalities, and applying Gronwall’s inequality to (3.9), one can easily get that
According to (3.10), we obtain
Integrating (3.11) on with , by virtue of (3.10), we get
If , for all , the above inequality implies that
Therefore, is uniformly bounded in . In view of Theorem 1.2, the solution can be extended beyond . This is in conflict with the definition of . □
Remark 3.1 If , , in view of being an algebra, we have (3.10). Thus we also deduce the result of Theorem 1.4.
Proof of Theorem 1.5 We now derive the multi-peakon solutions of Eq. (1.1). Assume that Eq. (1.1) has an N-peakon solution of the form (1.4). It follows from the definition of a weak solution that for any , the solution (1.1) satisfies
which is equivalent to the following equation:
where , .
A straightforward computation gives
Thus, combining (3.15) with (3.16), we get
Substituting (3.14), (3.17) into (3.13), we obtain the following system:
which leads to the conclusion of Theorem 1.5. □
4 Analysis of the Novikov equation in weighted spaces
In this section, for the convenience of the readers, we first present some standard definitions. In general, a weight function is simply a non-negative function. A weight function is called sub-multiplicative if
Given a sub-multiplicative function v, a positive function ϕ is v-moderate if and only if
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 . Let us recall the most standard examples of such weights. Let
We have (see ) the following conditions:
For and , such weight is sub-multiplicative.
If and , then ϕ is moderate. More precisely, is -moderate for , , and .
The elementary properties of sub-multiplicative and moderate weights can be found in . Next, we prove Theorem 1.6.
Proof of Theorem 1.6 We define
We also introduce the kernel . Then Eq. (1.1) can be rewritten as
Note that from the assumption , , we get
For any , let us consider the N-truncations
Observe that is a locally absolutely continuous function such that
In addition, if , where , then
Indeed, let us introduce the set , if , then ; if , then .
The constant being independent on N shows that the N-truncations of a v-moderate weight are uniformly v-moderate with respect to N.
We start considering the case . Multiplying Eq. (4.1) by f and then by , we get, after integration,
Note that the estimates
are true. Moreover, we get
In the first inequality we used Hölder’s inequality, and in the second inequality we applied Proposition 3.2 in , and in the last one we used condition (1.6). Here, C depends only on v and ϕ. From (4.2) we can obtain
Next, we give estimates on . Differentiating (4.1) with respect to x-variable, next multiplying by f produces the equation
Multiplying this equation by with , integrating the result in the x-variable, we note that
In the third inequality we applied the pointwise bound and the condition
In the last inequality we used for a.e. x. Thus, we get
Now, combing inequalities (4.3) and (4.4) and then integrating yields
Since as for a.e. . Recalling that and , we get
At last, we treat the case . We have and . Hence, we have
The last factor on the right-hand side is independent of q. Since as for any implies that
The last factor on the right-hand side is independent of N. Now taking implies that estimate (4.5) remains valid for . □
Proof of Theorem 1.7 We start observing that is a -moderate weight such that . Moreover, . By condition (1.7), , hence Hölder’s inequality implies that . Then Theorem 1.6 applied with to the weight yields
Arguing as in the proof of Theorem 1.6, we get
On the other hand,
Note that and
The constant on the right-hand side is dependent on N. Similarly, recalling that , we obtain
Plugging the two last estimates in (4.6)-(4.7) and summing, we obtain
Integrating and finally letting yields the conclusion in the case . The constants throughout the proof are independent on p. Therefore, for , one can rely on the result established for finite exponents q, and then let . The rest argument is fully similar to that of Theorem 1.6. □
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The authors are very grateful to the anonymous reviewers and editors for their careful read and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by NSFC grant 11301573 and in part by the Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308.
The authors declare that they have no competing interests.
This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.