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The Cauchy problem for the generalized Degasperis-Procesi equation
Boundary Value Problems volumeĀ 2013, ArticleĀ number:Ā 235 (2013)
Abstract
In this paper, we investigate the Cauchy problem for the generalized Degasperis-Procesi equation in a Besov space. Firstly, we prove that the generalized Degasperis-Procesi equation is locally well posed in with (or if with ). Secondly, we prove that the generalized Degasperis-Procesi equation possesses the peaked solitary wave which is the weak solution to the generalized Degasperis-Procesi equation. Thirdly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in . Fourthly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in with . Finally, we give a blow-up criterion.
MSC:35G25, 35L05.
1 Introduction
In this paper, we consider the Cauchy problem for the following generalized Degasperis-Procesi equation:
where is a constant. When and , (1.1) reduces to the Degasperis-Procesi equation
Equation (1.3) possesses the Lax pair and bi-Hamiltonian structures and infinite many conservation laws [1]. The Degasperis-Procesi equation [2] possesses peaked solitons which are stable [3] and shock peakons of the form , . The Degasperis-Procesi equation possesses the global weak solution and blow-up structure [4ā6]. Constantin and Lannes studied the relevance between the Camassa-Holm equation and the Degasperis-Procesi equation [7]. The Degasperis-Procesi equation possesses the infinite propagation speed [8]. The Degasperis-Procesi equation possesses multi-peakon solutions [9] and multisoliton [10]. Himonas and his co-authors [11, 12] proved that the data-to-solution for the Camassa-Holm equation, the Degasperis-Procesi equation is not uniformly continuous in with , respectively. Himonas et al. proved the non-uniform continuity in of the solution map of the CH equation [13]. Recently, Gui and Liu [14] studied the Cauchy problem for the Degasperis-Procesi equation in Besov spaces. Yan et al. [15] studied the Cauchy problem for the Novikov equation in Besov spaces.
Let and , . By using the identity for , we can rewrite (1.1)-(1.2) as follows:
In this paper, motivated by [14], we study the Cauchy problem for (1.4) in Besov spaces. Firstly, we use the standard iterative method to prove that the generalized Degasperis-Procesi equation is locally well posed in with (or if with ). Secondly, we prove that the generalized Degasperis-Procesi equation possesses the peaked solitary wave which is the weak solution to the generalized Degasperis-Procesi equation. Thirdly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in . Fourthly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in with . Finally, we give a blow-up criterion.
Notice that the structure of (1.4) is more complicated than that of the Degasperis-Procesi equation. Thus, to prove that the sequence of smooth solutions is uniformly bounded in with (or if with ), we choose that
In proving Theorem 1.5, we explain why we choose (1.6). It is worthy of pointing out that we use Fatouās lemma and the upper limit as well as Gronwallās inequality to prove that is a Cauchy sequence in with (or if with ).
To introduce the main results, we define
The main results of this paper are as follows.
Theorem 1.1 Let with (or if with ). Problem (1.4)-(1.5) is locally well posed. Moreover,
A function is called a weak solution to (1.1) (or (1.4)) if u belongs to and satisfies the following identity:
where , for any smooth test function . If u is a weak solution on for every , then it is called a global weak solution.
Theorem 1.2 When in (1.4), with is a weak solution of (1.4) in the sense of (1.8).
Theorem 1.3 When in (1.4), the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in . More precisely, there exists a global solution to the Cauchy problem for (1.4) such that for any and , there exists a solution with
Theorem 1.4 When in (1.4), the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in with .
Theorem 1.5 Assume that is the maximal time of existence of the solution to problem (1.4)-(1.5). If , then
Moreover, .
The remainder of this paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we prove Theorem 1.1. In Section 4, we prove Theorem 1.2. In Section 5, we prove Theorem 1.3. In Section 6, we prove Theorem 1.4. In Section 7, we prove Theorem 1.5.
2 Preliminaries
In this section, we give Lemmas 2.1-2.4. The proof of Lemmas 2.1-2.4 can be seen in [16ā21].
Lemma 2.1 (Littlewood-Paley decomposition)
Let and . There exists a couple of smooth radial functions such that
and
and
Then, for , the nonhomogeneous dyadic blocks are defined as follows:
Thus we obtain
and the low frequency cut-off is defined by
as well as
where C is a positive constant independent of q.
Definition (Besov spaces)
Let and . The nonhomogeneous Besov space is defined by
In particular, if , then .
Lemma 2.2 Let , , , then:
-
(1)
is a Banach space and is continuously embedded in .
-
(2)
, if and and
-
(3)
, is a Banach algebra. is a Banach algebra iff and iff or ( and ). (4)
-
(i)
For ,
-
(ii)
( if ) and ,
-
(5)
and ,
-
(6)
If is bounded in and in , then and
-
(7)
Let and ĪØ be an -multiplier. Then the operator is continuous from into . In particular, is continuous from into .
Lemma 2.3 (A priori estimates in Besov spaces)
Let and . Assume that , and that belongs to if or to otherwise. If solves the following 1-D linear transport equation:
then there exists a constant C depending only on s, p, r such that the following statements hold:
-
(1)
If or , then
or
with if and else.
-
(2)
If , and and , then
with
-
(3)
If , then for all , (1) holds true when .
-
(4)
If , then . If , then for all .
Lemma 2.4 (Existence and uniqueness)
Let p, r, s, and F be as in the statement of Lemma 2.3. Assume that for some and and if or and and if . Then problem (2.1)-(2.2) has a unique solution and the inequalities of Lemma 2.3 can hold true. Moreover, if , then .
3 Proof of Theorem 1.1
In this section, we complete the proof of Theorem 1.1 and suppose that (or if with ).
First step: approximate solution
By using the standard iterative process, we construct a sequence of smooth solutions . Assume that , by induction we define a sequence of smooth functions by solving the following linear transport equation:
By using the fact that belong to , from Lemma 2.4, for all , we can show by induction that problem (3.1)-(3.2) has a global solution .
Second step: uniform bounds
For (or if with ) and , we prove that
with .
From (2.3) of Lemma 2.3 and (3.1), we derive that
where
By using the multiplier property of and the fact with (or if with ) is a Banach algebra, we have
Inserting (3.6) into (3.4) yields (3.3).
Let us fix such that
and suppose that
Since , by using (3.8), we have
When in (3.9), we have
Inserting (3.9), (3.10) into (3.3) yields
Thus, is uniformly bounded in . By using (3.9) and the fact that is a Banach algebra and (3.5) and using the -multiplier property of , we have that
and
Consequently,
Remark Inserting (3.7) into (3.8) yields
for . From (3.11) and (3.7), , we have that
and
with the aid of Fatouās lemma. We define
Thus,
for .
Third step: convergence
We prove that is a Cauchy sequence in . For , we have
Combining (2.3) with (3.20), we have that
where
By using (3) and (7) of Lemma 2.2, we have that
Inserting (3.22) into (3.21) yields
From (3.2) and Lemma 2.1, we can easily obtain that
Obviously,
Inserting (3.24) and (3.25) into (3.23) leads to
We define
Inserting (3.27) into (3.26) leads to
We define
and
Combining (3.30) with (3.29), (3.28), by using Fatouās lemma, we have that
Applying Gronwallās inequality to (3.31) yields
for . According to the definition of , we can easily obtain that
Combining (3.32) with (3.33), we have that
Hence, is a Cauchy sequence in .
Fourth step: existence in
Now we prove that and satisfies (1.4)-(1.5) since is uniformly bounded in . From (6) in Lemma 2.2, we have that . From (1.4), we can easily prove that . It is easily checked that u is indeed a solution to (1.4)-(1.5) by passing to the limit in (3.1)-(3.2).
Now we prove that . Since , , there exists such that
From the definition of Besov spaces, we have that
By using the mean value theorem, we have that
where . Inserting (3.37) into (3.36) yields
We may choose Ī“ sufficiently small such that
Inserting (3.39) into (3.38) leads to
Thus, we derive that
Combining (3.41) with (1.4), we can easily obtain
From (3.41) and (3.42), we have that .
Fifth step: uniqueness of solution
The uniqueness of the solution to the Cauchy problem for (1.4) can be proved similarly to Proposition 3.1 of [14].
Sixth step: continuity in with (or if with )
The continuity of the solution to the Cauchy problem for (1.4) can be proved similarly to the continuity of the solution to the Degasperis-Procesi equation which can be seen in [14].
4 Proof of Theorem 1.2
From (6.5) and (6.7) of [22], we have that
where . By using integration by parts and (4.1), we have that
Since , we have that when ,
and when ,
By using and (4.1), we have that
Now, we compute
When , we have that
and
and
Thus, when , from (4.7)-(4.9), we have that
Similarly, when , we can obtain
From (4.3), (4.4) and (4.10) as well as (4.11), we have that
Thus, is the solution in the sense of (4.12). For , let . Thus is the solitary wave for (1.1) (or (1.4)).
5 Proof of Theorem 1.3
Since when in (1.4), (1.4) possesses the peaked solitary wave , Theorem 1.3 can be proved similarly to Proposition 4 of [19].
6 Proof of Theorem 1.4
Since when in (1.4), (1.4) possesses the peaked solitary wave , Theorem 1.4 can be proved similarly to Theorem 3 of [23].
7 Proof of Theorem 1.5
In this section, we always assume that (or if with ).
Proof of Theorem 1.5 Applying to (1.4) yields
From (2.54) of page 112 in [24], since (or if with ), we have that
By using (4) of Lemma 2.2 and is an -multiplier, since (or if with ), we have that
Going along the lines of the proof of Proposition A.1 of [18], from (7.2) and (7.3), we have that
Solving (7.4) yields
Solving (7.5) yields
Assume that is the maximal time of existence of the solution to problem (1.4)-(1.5). If , we claim that
We prove the claim (7.8) by contradiction. If (7.8) is untrue, then from (7.8), we have that
which contradicts the fact that is the maximal time of existence of the solution to problem (1.4)-(1.5). Consequently, (7.8) is true. From (7.7), we know that . Moreover, (7.7) ensures the validity of (3.8).
The proof of Theorem 1.5 is completed.āā”
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Acknowledgements
We wish to thank the referee for a careful reading and valuable comments on the original draft. The first author is supported by NNSFC under grant No.Ā 11226185. The third author is supported by Foundation and Frontier of Henan Province under grant Nos.Ā 122300410414, 132300410432.
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Zuo, F., Tian, C. & Wang, H. The Cauchy problem for the generalized Degasperis-Procesi equation. Bound Value Probl 2013, 235 (2013). https://doi.org/10.1186/1687-2770-2013-235
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DOI: https://doi.org/10.1186/1687-2770-2013-235