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The Cauchy problem for the generalized Degasperis-Procesi equation

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 B p , r s with s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)). 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  B 2 , āˆž 3 / 2 . Fourthly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in H s (R) with s<3/2. 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:

u t āˆ’ u t x x + u k u x āˆ’ ( u k u x ) x x +Q [ u k + 1 ] x =0,
(1.1)
u(x,0)= u 0 (x),
(1.2)

where QāˆˆR is a constant. When k=1 and Q= 3 2 , (1.1) reduces to the Degasperis-Procesi equation

u t āˆ’ u t x x +4u u x =3 u x u x x +u u x x x .
(1.3)

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 u(x,t)=āˆ’ 1 t + k sign(x) e āˆ’ | x | , k>0. The Degasperis-Procesi equation possesses the global weak solution and blow-up structure [4ā€“6]. 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 H s (R) with s>3/2, respectively. Himonas et al. proved the non-uniform continuity in H 1 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 P(D)=āˆ’ āˆ‚ x ( 1 āˆ’ āˆ‚ x 2 ) āˆ’ 1 and p(x)= 1 2 e āˆ’ | x | , xāˆˆR. By using the identity ( 1 āˆ’ āˆ‚ x 2 ) āˆ’ 1 f=pāˆ—f for fāˆˆ L 2 , we can rewrite (1.1)-(1.2) as follows:

u t + u k u x =QP(D) [ u k + 1 ] ,
(1.4)
u(x,0)= u 0 (x).
(1.5)

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 B p , r s with s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)). 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 B 2 , āˆž 3 / 2 . Fourthly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in H s (R) with s<3/2. 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 ( u ( n ) ) n āˆˆ N is uniformly bounded in C([0,T]; B p , r s )āˆ© C 1 ([0,T]; B p , r s āˆ’ 1 ) with s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)), we choose that

āˆ„ u ( n ) āˆ„ B p , r s ā‰¤ āˆ„ u 0 āˆ„ B p , r s ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 / k ,tāˆˆ[0,T].
(1.6)

In proving Theorem 1.5, we explain why we choose (1.6). It is worthy of pointing out that we use Fatouā€™s lemma and the upper limit as well as Gronwallā€™s inequality to prove that ( u ( n ) ) n āˆˆ N is a Cauchy sequence in C([0,T]; B p , r s āˆ’ 1 ) with s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)).

To introduce the main results, we define

E p , r s (T)=C ( [ 0 , T ] ; B p , r s ) āˆ© C 1 ( [ 0 , T ] ; B p , r s āˆ’ 1 ) .

The main results of this paper are as follows.

Theorem 1.1 Let u 0 (x)āˆˆ B p , r s with s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)). Problem (1.4)-(1.5) is locally well posed. Moreover,

āˆ„ u ( t ) āˆ„ B p , r s ā‰¤ āˆ„ u 0 āˆ„ B p , r s ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 / k ,tāˆˆ[0,T].
(1.7)

A function u:[0,T)ƗR is called a weak solution to (1.1) (or (1.4)) if u belongs to L loc āˆž ([0,T); H 1 ) and satisfies the following identity:

āˆ« 0 T āˆ« R [ u Ļ• t + 1 k + 1 u k + 1 Ļ• x + p āˆ— [ k 2 + 2 k k + 1 u k + 1 ] Ļ• x ] d x d t + āˆ« R u ( x , 0 ) Ļ• ( x , 0 ) d x = 0 ,
(1.8)

where p(x)= 1 2 e āˆ’ | x | , for any smooth test function Ļ•(x,t)āˆˆ C c āˆž ([0,T)ƗR). If u is a weak solution on [0,T) for every T>0, then it is called a global weak solution.

Theorem 1.2 When Q= k ( k + 2 ) k + 1 in (1.4), u c (x,t)= c 1 / k e āˆ’ | x āˆ’ c t | with c>0 is a weak solution of (1.4) in the sense of (1.8).

Theorem 1.3 When Q= k ( k + 2 ) k + 1 in (1.4), the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in B 2 , āˆž 3 / 2 . More precisely, there exists a global solution uāˆˆ L āˆž ( R + ; B 2 , āˆž 3 / 2 ) to the Cauchy problem for (1.4) such that for any T>0 and Ļµ>0, there exists a solution vāˆˆ L āˆž (0,T; B 2 , āˆž 3 / 2 ) with

āˆ„ v ( 0 ) āˆ’ u ( 0 ) āˆ„ B 2 , āˆž 3 / 2 ā‰¤Ļµ, āˆ„ v ( t ) āˆ’ u ( t ) āˆ„ L āˆž ( 0 , T ; B 2 , āˆž 3 / 2 ) ā‰„1.

Theorem 1.4 When Q= k ( k + 2 ) k + 1 in (1.4), the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in H s (R) with s<3/2.

Theorem 1.5 Assume that T ā‹† is the maximal time of existence of the solution to problem (1.4)-(1.5). If T ā‹† <āˆž, then

āˆ« 0 T ā‹† āˆ„ u x āˆ„ L āˆž k dĻ„=+āˆž.
(1.9)

Moreover, T ā‹† ā‰„ 1 C k āˆ„ u 0 āˆ„ B p , r s k .

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 [16ā€“21].

Lemma 2.1 (Littlewood-Paley decomposition)

Let B={Ī¾āˆˆ R n ,|Ī¾|ā‰¤ 4 3 } and C={Ī¾āˆˆ R n , 3 4 ā‰¤|Ī¾|ā‰¤ 8 3 }. There exists a couple of smooth radial functions (Ļ‡,Ļ•)āˆˆ( C c āˆž (B), C c āˆž (C)) such that

āˆ€Ī¾āˆˆ R n ,Ļ‡(Ī¾)+ āˆ‘ q āˆˆ N Ļ• ( 2 āˆ’ q Ī¾ ) =1

and

Supp Ļ• ( 2 āˆ’ q ā‹… ) āˆ© Supp Ļ• ( 2 āˆ’ q ā€² ā‹… ) = āˆ… if | q āˆ’ q ā€² | ā‰„ 2 , Supp Ļ‡ ( ā‹… ) āˆ© Supp Ļ• ( 2 āˆ’ q ā‹… ) = āˆ… if | q | ā‰„ 1

and

1 3 ā‰¤Ļ‡ ( Ī¾ ) 2 + āˆ‘ q ā‰„ 0 Ļ• ( 2 āˆ’ q Ī¾ ) 2 ā‰¤1,āˆ€Ī¾āˆˆ R n .
(2.1)

Then, for uāˆˆ S ā€² (R), the nonhomogeneous dyadic blocks are defined as follows:

Ī” q u = 0 if q ā‰¤ āˆ’ 2 , Ī” āˆ’ 1 u = Ļ‡ ( D ) u = F x āˆ’ 1 Ļ‡ F x u , Ī” q u = Ļ• ( 2 āˆ’ q D ) = F x āˆ’ 1 Ļ• ( 2 āˆ’ q Ī¾ ) F x u if q ā‰„ 0 .

Thus we obtain

u= āˆ‘ q āˆˆ Z Ī” q uin S ā€² (R),

and the low frequency cut-off S q is defined by

S q u= āˆ‘ p = āˆ’ 1 q āˆ’ 1 Ī” p u=Ļ‡ ( 2 āˆ’ q D ) u= F x āˆ’ 1 Ļ‡ ( 2 āˆ’ q Ī¾ ) F x u,āˆ€qāˆˆN,

as well as

Ī” p Ī” q u ā‰” 0 if | p āˆ’ q | ā‰„ 2 , Ī” q ( S p āˆ’ 1 u Ī” p v ) ā‰” 0 if | p āˆ’ q | ā‰„ 5 , āˆ€ u , v āˆˆ S ā€² ( R ) , āˆ„ Ī” p u āˆ„ L p ā‰¤ C āˆ„ u āˆ„ L p , āˆ„ S q u āˆ„ L p ā‰¤ C āˆ„ u āˆ„ L p , āˆ€ 1 ā‰¤ p ā‰¤ + āˆž ,

where C is a positive constant independent of q.

Definition (Besov spaces)

Let sāˆˆR and 1ā‰¤pā‰¤+āˆž. The nonhomogeneous Besov space B p , r s ( R n ) is defined by

B p , r s ( R n ) = { f āˆˆ S ā€² ( R n ) : āˆ„ f āˆ„ B p , r s = āˆ„ 2 q s Ī” q f āˆ„ l r ( L p ) = āˆ„ ( 2 q s āˆ„ Ī” q f āˆ„ L p ) q ā‰„ āˆ’ 1 āˆ„ l r < āˆž } .

In particular, if s=āˆž, then B p , r s = ā‹‚ s āˆˆ R B p , r s .

Lemma 2.2 Let sāˆˆR, 1ā‰¤p,r, p j , r j ā‰¤āˆž, j=1,2, then:

  1. (1)

    B p , r s is a Banach space and is continuously embedded in S ā€² ( R n ).

  2. (2)

    B p 1 , r 1 s 1 ā†Ŗ B p 2 , r 2 s 2 , if p 1 ā‰¤ p 2 and r 1 ā‰¤ r 2 and s 2 = s 1 āˆ’n( 1 p 1 āˆ’ 1 p 2 )

    B p , r 2 s 1 ā†Ŗ B p , r 1 s 2 locally compact if s 2 < s 1 .
  3. (3)

    āˆ€s>0, B p , r s āˆ© L āˆž is a Banach algebra. B p , r s is a Banach algebra iff B p , r s ā†Ŗ L āˆž and iff s> 1 p or (sā‰„ 1 p and r=1). (4)

  4. (i)

    For s>0,

    āˆ„ f g āˆ„ B p , r s ā‰¤C ( āˆ„ f āˆ„ B p , r s āˆ„ g āˆ„ L āˆž + āˆ„ f āˆ„ L āˆž āˆ„ g āˆ„ B p , r s ) ,āˆ€f,gāˆˆ B p , r s āˆ© L āˆž .
  5. (ii)

    āˆ€ s 1 ā‰¤ 1 p < s 2 ( s 2 ā‰„ 1 p if r=1) and s 1 + s 2 >0,

    āˆ„ f g āˆ„ B p , r s 1 ā‰¤C āˆ„ f āˆ„ B p , r s 1 āˆ„ g āˆ„ B p , r s 2 ,āˆ€fāˆˆ B p , r s 1 ,gāˆˆ B p , r s 2 .
  6. (5)

    āˆ€Īøāˆˆ[0,1] and s=Īø s 1 +(1āˆ’Īø) s 2 ,

    āˆ„ f āˆ„ B p , r s ā‰¤C āˆ„ f āˆ„ B p , r s 1 Īø āˆ„ f āˆ„ B p , r s 2 1 āˆ’ Īø ,āˆ€fāˆˆ B p , r s 1 āˆ© B p , r s 2 .
  7. (6)

    If ( u n ) n āˆˆ N is bounded in B p , r s and u n ā†’u in S ā€² ( R n ), then uāˆˆ B p , r s and

    āˆ„ u āˆ„ B p , r s ā‰¤ limā€‰inf n ā†’ āˆž āˆ„ u n āˆ„ B p , r s .
  8. (7)

    Let māˆˆR and ĪØ be an S m -multiplier. Then the operator ĪØ(D) is continuous from B p , r s into B p , r s āˆ’ m . In particular, āˆ’ āˆ‚ x ( 1 āˆ’ āˆ‚ x 2 ) āˆ’ 1 is continuous from B p , r s into B p , r s āˆ’ 1 .

Lemma 2.3 (A priori estimates in Besov spaces)

Let 1ā‰¤p,rā‰¤āˆž and s>āˆ’min{ 1 p ,1āˆ’ 1 p }. Assume that f 0 āˆˆ B p , r s , Fāˆˆ L 1 (0,T; B p , r s ) and that āˆ‚ x v belongs to L 1 (0,T; B p , r s āˆ’ 1 ) if s>1+ 1 p or to L 1 (0,T; B p , r 1 / p āˆ© L āˆž ) otherwise. If fāˆˆ L āˆž (0,T; B p , r s )āˆ©C([0,T]; S ā€² (R)) solves the following 1-D linear transport equation:

f t +v f x =F,
(2.2)
f(x,0)= f 0 ,
(2.3)

then there exists a constant C depending only on s, p, r such that the following statements hold:

  1. (1)

    If r=1 or sā‰ 1+ 1 p , then

    āˆ„ f āˆ„ B p , r s ā‰¤ āˆ„ f 0 āˆ„ B p , r s + āˆ« 0 t āˆ„ F ( Ļ„ ) āˆ„ B p , r s dĻ„+C āˆ« 0 t V ā€² (Ļ„) āˆ„ f ( Ļ„ ) āˆ„ B p , r s dĻ„

or

āˆ„ f āˆ„ B p , r s ā‰¤ e C V ( t ) ( āˆ„ f 0 āˆ„ B p , r s + āˆ« 0 t e āˆ’ C V ( Ļ„ ) āˆ„ F ( Ļ„ ) āˆ„ B p , r s d Ļ„ )
(2.4)

with V(t)= āˆ« 0 t āˆ„ v x ( Ļ„ ) āˆ„ B p , r 1 / p āˆ© L āˆž dĻ„ if s<1+ 1 p and V(t)= āˆ« 0 t āˆ„ v x ( Ļ„ ) āˆ„ B p , r s āˆ’ 1 dĻ„ else.

  1. (2)

    If sā‰¤1+ 1 p , f 0 ā€² āˆˆ L āˆž and f x āˆˆ L āˆž ((0,T)ƗR) and F x āˆˆ L 1 (0,T; L āˆž ), then

    āˆ„ f ( t ) āˆ„ B p , r s + āˆ„ f x ( t ) āˆ„ L āˆž ā‰¤ e C V ( t ) ( āˆ„ f 0 āˆ„ B p , r s + āˆ„ f 0 ā€² āˆ„ L āˆž + āˆ« 0 t e āˆ’ C V ( Ļ„ ) [ āˆ„ F ( Ļ„ ) āˆ„ B p , r s + āˆ„ F x ( Ļ„ ) āˆ„ L āˆž ] d Ļ„ )

with

V(t)= āˆ« 0 t āˆ„ āˆ‚ x v ( Ļ„ ) āˆ„ B p , r 1 / p āˆ© L āˆž dĻ„.
  1. (3)

    If f=v, then for all s>0, (1) holds true when V(t)= āˆ« 0 t āˆ„ v x ( Ļ„ ) āˆ„ L āˆž dĻ„.

  2. (4)

    If r<āˆž, then fāˆˆC([0,T]; B p , r s ). If r=āˆž, then fāˆˆC([0,T]; B p , 1 s ā€² ) for all s ā€² <s.

Lemma 2.4 (Existence and uniqueness)

Let p, r, s, f 0 and F be as in the statement of Lemma  2.3. Assume that vāˆˆ L Ļ (0,T; B āˆž , āˆž āˆ’ M ) for some Ļ>1 and M>0 and v x āˆˆ L 1 (0,T; B p , r s āˆ’ 1 ) if s>1+ 1 p or s=1+ 1 p and r=1 and v x āˆˆ L 1 (0,T; B p , āˆž 1 / p āˆ© L āˆž ) if s<1+ 1 p . Then problem (2.1)-(2.2) has a unique solution fāˆˆ L āˆž (0,T; B p , r s )āˆ©( ā‹‚ s ā€² < s C([0,T]; B p , 1 s ā€² )) and the inequalities of Lemma  2.3 can hold true. Moreover, if r<āˆž, then fāˆˆC([0,T]; B p , r s ).

3 Proof of Theorem 1.1

In this section, we complete the proof of Theorem 1.1 and suppose that s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)).

First step: approximate solution

By using the standard iterative process, we construct a sequence of smooth solutions ( u ( n ) ) n āˆˆ N āˆˆC( R + ; B p , r āˆž ). Assume that u ( 0 ) :=0, by induction we define a sequence of smooth functions ( u ( n ) ) n āˆˆ N by solving the following linear transport equation:

u t ( n + 1 ) + [ u ( n ) ] k u x ( n + 1 ) =QP(D) [ ( u ( n ) ) k + 1 ] ,
(3.1)
u ( n + 1 ) (x,0)= u 0 ( n + 1 ) (x)= S n + 1 u 0 (x).
(3.2)

By using the fact that S n + 1 u 0 belong to B p , r āˆž , from Lemma 2.4, for all nāˆˆN, we can show by induction that problem (3.1)-(3.2) has a global solution ( u ( n ) ) n āˆˆ N āˆˆC( R + , B p , r āˆž ).

Second step: uniform bounds

For s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)) and nāˆˆN, we prove that

āˆ„ u ( n + 1 ) āˆ„ B p , r s ā‰¤ e C U n ( t ) ( āˆ„ u 0 āˆ„ B p , r s + C āˆ« 0 t e āˆ’ C U n ( Ļ„ ) āˆ„ u ( n ) āˆ„ B p , r s k + 1 d Ļ„ ) ,
(3.3)

with U n = āˆ« 0 t āˆ„ u ( n ) āˆ„ B p , r s k dĻ„.

From (2.3) of Lemma 2.3 and (3.1), we derive that

āˆ„ u ( n + 1 ) ( t ) āˆ„ B p , r s ā‰¤ e C āˆ« 0 t āˆ„ u x ( n ) ( t ā€² ) āˆ„ B p , r s āˆ’ 1 d t ā€² āˆ„ u 0 āˆ„ B p , r s + C āˆ« 0 t e C āˆ« Ļ„ t āˆ„ u x ( n ) ( t ā€² ) āˆ„ B p , r s āˆ’ 1 d t ā€² āˆ„ F ( u ( n ) ) āˆ„ B p , r s d Ļ„ ,
(3.4)

where

F ( u ( n ) ) =P(D) [ ( u ( n ) ) k + 1 ] .
(3.5)

By using the S āˆ’ 1 multiplier property of P(D) and the fact B p , r s with s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)) is a Banach algebra, we have

āˆ„ F ( u ( n ) ) āˆ„ B p , r s āˆ’ 1 ā‰¤ C āˆ„ P ( D ) [ ( u ( n ) ) k + 1 ] ( t ā€² ) āˆ„ B p , r s ā‰¤ C āˆ„ ( u ( n ) ) k + 1 ( t ā€² ) āˆ„ B p , r s āˆ’ 1 ā‰¤ C āˆ„ u ( n ) ( t ā€² ) āˆ„ B p , r s k + 1 .
(3.6)

Inserting (3.6) into (3.4) yields (3.3).

Let us fix T>0 such that

Tā‰¤ 1 4 k C āˆ„ u 0 āˆ„ B p , r s k ,
(3.7)

and suppose that

āˆ„ u ( n ) ( t ) āˆ„ B p , r s ā‰¤ āˆ„ u 0 āˆ„ B p , r s ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 / k ,tāˆˆ[0,T].
(3.8)

Since U n (t)= āˆ« 0 t āˆ„ u ( n ) āˆ„ B p , r s k dĻ„, by using (3.8), we have

e C ( U n ( t ) āˆ’ U n ( Ļ„ ) ) = e C āˆ« Ļ„ t āˆ„ u ( n ) ( t ā€² ) āˆ„ B p , r s k d t ā€² ā‰¤ e āˆ’ 1 2 k āˆ« Ļ„ t d ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ā€² ) ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ā€² ) = ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k Ļ„ 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 2 k .
(3.9)

When Ļ„=0 in (3.9), we have

e C U n ( t ) ā‰¤ ( 1 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 2 k .
(3.10)

Inserting (3.9), (3.10) into (3.3) yields

āˆ„ u ( n + 1 ) ( t ) āˆ„ B p , r s ā‰¤ āˆ„ u 0 āˆ„ B p , r s ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 / 2 k [ 1 āˆ’ 1 2 k āˆ« 0 t d ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 + 1 2 k ] ā‰¤ āˆ„ u 0 āˆ„ B p , r s ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 / k .
(3.11)

Thus, ( u ( n ) ) n āˆˆ N is uniformly bounded in C([0,T]; B p , r s ). By using (3.9) and the fact that B p , r s is a Banach algebra and (3.5) and using the S āˆ’ 1 -multiplier property of P(D), we have that

āˆ„ [ u ( n ) ] k u x ( n + 1 ) āˆ„ B p , r s ā‰¤C āˆ„ u ( n ) āˆ„ B p , r s k āˆ„ u ( n + 1 ) āˆ„ B p , r s ā‰¤ C āˆ„ u 0 āˆ„ B p , r s k + 1 ( 1 āˆ’ 2 C āˆ„ u 0 āˆ„ B p , r s k t ) k + 1 k
(3.12)

and

āˆ„ P ( D ) [ ( u ( n ) ) k + 1 ] āˆ„ B p , r s āˆ’ 1 ā‰¤ C āˆ„ ( u ( n ) ) k + 1 āˆ„ B p , r s āˆ’ 1 ā‰¤ C āˆ„ u ( n ) āˆ„ B p , r s k + 1 ā‰¤ C āˆ„ u 0 āˆ„ B p , r s k + 1 ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s t ) k + 1 k .
(3.13)

Consequently,

( u ( n ) ) n āŠ‚C ( [ 0 , T ] ; B p , r s ) āˆ© C 1 ( [ 0 , T ] ; B p , r s āˆ’ 1 ) .
(3.14)

Remark Inserting (3.7) into (3.8) yields

āˆ„ u ( n ) āˆ„ B p , r s ā‰¤4 āˆ„ u 0 āˆ„ B p , r s
(3.15)

for nāˆˆN. From (3.11) and (3.7), āˆ€nāˆˆ N + , we have that

e C U n ā‰¤exp [ āˆ« 0 t C āˆ„ u 0 āˆ„ B p , r s k 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t d Ļ„ ] ā‰¤4
(3.16)

and

āˆ„ u ( t ) āˆ„ B p , r s ā‰¤ āˆ„ u 0 āˆ„ B p , r s ( 1 āˆ’ 2 k C āˆ„ u 0 āˆ„ B p , r s k t ) 1 / k
(3.17)

with the aid of Fatouā€™s lemma. We define

L=4 ( āˆ„ u 0 āˆ„ B p , r s + 1 ) .
(3.18)

Thus,

āˆ„ u ( n ) āˆ„ B p , r s +1ā‰¤L
(3.19)

for nāˆˆ N + .

Third step: convergence

We prove that ( u ( n ) ) n āˆˆ N is a Cauchy sequence in C([0,T]; B p , r s āˆ’ 1 ). For (m,n)āˆˆ N 2 , we have

[ āˆ‚ t + ( u ( n + m ) ) k āˆ‚ x ] ( u ( n + 1 + m ) āˆ’ u ( n + 1 ) ) = ( ( u ( n ) ) k āˆ’ ( u ( n + m ) ) k ) āˆ‚ x u ( n + 1 ) + Q P ( D ) [ ( u ( n + m ) āˆ’ u ( n ) ) āˆ‘ j = 0 k ( u ( n + m ) ) k āˆ’ j ( u ( n ) ) j ] .
(3.20)

Combining (2.3) with (3.20), we have that

āˆ„ [ u ( n + 1 + m ) āˆ’ u ( n + 1 ) ] ( t ) āˆ„ B p , r s āˆ’ 1 ā‰¤ e U n ( t ) ( āˆ„ u 0 ( n + 1 + m ) āˆ’ u 0 ( n + 1 ) āˆ„ B p , r s āˆ’ 1 ) + āˆ« 0 t e U n ( t ) āˆ’ U n ( Ļ„ ) āˆ„ F ( u ( n ) , u ( n + m ) , āˆ‚ x u ( n + 1 ) ) āˆ„ B p , r s āˆ’ 1 d Ļ„ ,
(3.21)

where

U n ( t ) = āˆ« 0 t āˆ„ ( u ( n + m ) ) k āˆ„ B p , r s d Ļ„ , F ( u ( n ) , u ( n + m ) , āˆ‚ x u ( n + 1 ) ) = ( ( u ( n ) ) k āˆ’ ( u ( n + m ) ) k ) āˆ‚ x u ( n + 1 ) + Q P ( D ) [ ( u ( n + m ) āˆ’ u ( n ) ) āˆ‘ j = 0 k ( u ( n + m ) ) k āˆ’ j ( u ( n ) ) j ] .

By using (3) and (7) of Lemma 2.2, we have that

āˆ„ F ( u ( n ) , u ( n + m ) , āˆ‚ x u ( n + 1 ) ) āˆ„ B p , r s āˆ’ 1 ā‰¤ C āˆ„ ( u ( n ) ) k āˆ’ ( u ( n + m ) ) k āˆ„ B p , r s āˆ’ 1 āˆ„ āˆ‚ x u ( n + 1 ) āˆ„ B p , r s āˆ’ 1 + C āˆ„ P ( D ) [ ( u ( n + m ) āˆ’ u ( n ) ) āˆ‘ j = 0 k ( u ( n + m ) ) k āˆ’ j ( u ( n ) ) j ] āˆ„ B p , r s āˆ’ 1 ā‰¤ C āˆ„ u ( n ) āˆ’ u ( n + m ) āˆ„ B p , r s āˆ’ 1 [ āˆ„ u ( n ) āˆ„ B p , r s + āˆ„ u ( n + 1 ) āˆ„ B p , r s + āˆ„ u ( n + m ) āˆ„ B p , r s + 1 ] k ā‰¤ C L k āˆ„ u ( n ) āˆ’ u ( n + m ) āˆ„ B p , r s āˆ’ 1 .
(3.22)

Inserting (3.22) into (3.21) yields

āˆ„ [ u ( n + 1 + m ) āˆ’ u ( n + 1 ) ] ( t ) āˆ„ B p , r s āˆ’ 1 ā‰¤ e U n ( t ) ( āˆ„ u 0 ( n + 1 + m ) āˆ’ u 0 ( n + 1 ) āˆ„ B p , r s āˆ’ 1 ) + C L k āˆ« 0 t e U n ( t ) āˆ’ U n ( Ļ„ ) āˆ„ u ( n ) āˆ’ u ( n + m ) āˆ„ B p , r s āˆ’ 1 d Ļ„ .
(3.23)

From (3.2) and Lemma 2.1, we can easily obtain that

āˆ„ u 0 ( n + 1 + m ) āˆ’ u 0 ( n + 1 ) āˆ„ B p , r s āˆ’ 1 ā‰¤C 2 āˆ’ n .
(3.24)

Obviously,

e U n ( t ) ā‰¤4, e U n ( t ) āˆ’ U n ( Ļ„ ) ā‰¤4.
(3.25)

Inserting (3.24) and (3.25) into (3.23) leads to

āˆ„ ( u ( n + 1 + m ) āˆ’ u ( n + 1 ) ) ( t ) āˆ„ B p , r s āˆ’ 1 ā‰¤ C 2 āˆ’ n + C L k āˆ« 0 t āˆ„ ( u ( n ) āˆ’ u ( n + m ) ) ( Ļ„ ) āˆ„ B p , r s āˆ’ 1 d Ļ„ .
(3.26)

We define

A ( n , m ) (t)= āˆ„ ( u ( n + m ) āˆ’ u ( n ) ) āˆ„ B p , r s āˆ’ 1 .
(3.27)

Inserting (3.27) into (3.26) leads to

A ( n + 1 , m ) (t)ā‰¤C 2 āˆ’ n +C L k āˆ« 0 t A ( n , m ) (Ļ„)dĻ„.
(3.28)

We define

Ļ n (t)= sup m āˆˆ N + A ( n , m ) (t)= sup m āˆˆ N + āˆ„ ( u ( n + m ) āˆ’ u ( n ) ) ( t ) āˆ„ B p , r s
(3.29)

and

Ļ Ėœ (t)= limā€‰sup n ā†’ + āˆž Ļ n (t).
(3.30)

Combining (3.30) with (3.29), (3.28), by using Fatouā€™s lemma, we have that

Ļ Ėœ (t)= limā€‰sup n ā†’ + āˆž Ļ n + 1 (t)ā‰¤ C k āˆ« 0 t Ļ Ėœ (Ļ„)dĻ„.
(3.31)

Applying Gronwallā€™s inequality to (3.31) yields

Ļ Ėœ (t)ā‰¤ e t C k Ļ Ėœ (0)
(3.32)

for tāˆˆ[0,T]. According to the definition of Ļ Ėœ (t), we can easily obtain that

Ļ Ėœ (0)=0.
(3.33)

Combining (3.32) with (3.33), we have that

Ļ Ėœ (t)=0.
(3.34)

Hence, ( u n ) n is a Cauchy sequence in C([0,T]; B p , r s āˆ’ 1 ).

Fourth step: existence in E p , r s (T)

Now we prove that uāˆˆ E p , r s (T) and satisfies (1.4)-(1.5) since ( u n ) n āˆˆ N is uniformly bounded in L āˆž (0,T; B p , r s ). From (6) in Lemma 2.2, we have that uāˆˆ L āˆž (0,T; B p , r s ). From (1.4), we can easily prove that u t āˆˆ L āˆž (0,T; B p , r s āˆ’ 1 ). 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 uāˆˆ E p , r s (T). Since uāˆˆ B p , r s , āˆ€Ļµ>0, there exists q 0 āˆˆ N + such that

āˆ‘ q ā‰„ q 0 2 q s r āˆ„ Ī” q u āˆ„ L āˆž ( 0 , T ; L p ) r ā‰¤ Ļµ 4 .
(3.35)

From the definition of Besov spaces, we have that

āˆ„ u ( t + Ī“ ) āˆ’ u ( t ) āˆ„ B p , r s r = āˆ‘ q < q 0 2 q s r āˆ„ Ī” q u ( t ) āˆ’ Ī” q u ( t + Ī“ ) āˆ„ L p r + āˆ‘ q ā‰„ q 0 2 q s r āˆ„ Ī” q u āˆ’ Ī” q u ( t + Ī“ ) āˆ„ L p r ā‰¤ āˆ‘ q < q 0 2 q s r āˆ„ Ī” q u ( t ) āˆ’ Ī” q u ( t + Ī“ ) āˆ„ L p r + 2 āˆ‘ q ā‰„ q 0 2 q s r āˆ„ Ī” q u ( t ) āˆ„ L āˆž ( 0 , T ; L p ) r ā‰¤ āˆ‘ q < q 0 2 q s r āˆ„ Ī” q u ( t ) āˆ’ Ī” q u ( t + Ī“ ) āˆ„ L p r + Ļµ 2 .
(3.36)

By using the mean value theorem, we have that

āˆ„ Ī” q u ( t ) āˆ’ Ī” q u ( t + Ī“ ) āˆ„ L p = āˆ„ Ī” q u t ( t + Īø Ī“ ) āˆ„ L p |Ī“|ā‰¤ āˆ„ Ī” q u t ( t ) āˆ„ L āˆž ( 0 , T ; L p ) |Ī“|,
(3.37)

where 0<Īø<1. Inserting (3.37) into (3.36) yields

āˆ„ u ( t + Ī“ ) āˆ’ u ( t ) āˆ„ B p , r s r = āˆ‘ q < q 0 2 q s r āˆ„ Ī” q u ( t ) āˆ’ Ī” q u ( t + Ī“ ) āˆ„ L p r + āˆ‘ q ā‰„ q 0 2 q s r āˆ„ Ī” q u ( t ) āˆ’ Ī” q u ( t + Ī“ ) āˆ„ L p r ā‰¤ | Ī“ | āˆ‘ q < q 0 āˆ„ Ī” q u t ( t ) āˆ„ L āˆž ( 0 , T ; L p ) r + Ļµ 2 ā‰¤ | Ī“ | āˆ„ u t āˆ„ L āˆž ( 0 , T ; B p , r s āˆ’ 1 ) r + Ļµ 2 .
(3.38)

We may choose Ī“ sufficiently small such that

|Ī“| āˆ„ u t āˆ„ L āˆž ( 0 , T ; B p , r s āˆ’ 1 ) r ā‰¤ Ļµ 2 .
(3.39)

Inserting (3.39) into (3.38) leads to

āˆ„ u ( t + Ī“ ) āˆ’ u ( t ) āˆ„ B p , r s ā‰¤ Ļµ 1 / r .
(3.40)

Thus, we derive that

uāˆˆC ( [ 0 , T ] ; B p , r s ) .
(3.41)

Combining (3.41) with (1.4), we can easily obtain

u t āˆˆC ( [ 0 , T ] ; B p , r s āˆ’ 1 ) .
(3.42)

From (3.41) and (3.42), we have that uāˆˆ E p , r s (T).

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 E p , r s (T) with s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž))

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

āˆ‚ x u c (x,t)=āˆ’sign(xāˆ’ct) u c (x,t), āˆ‚ t u c (x,t)=csign(xāˆ’ct) u c (x,t),
(4.1)

where u c (x,t)= c 1 / k e āˆ’ | x āˆ’ c t | . By using integration by parts and (4.1), we have that

āˆ« 0 + āˆž āˆ« R ( u c āˆ‚ t Ļ• + 1 k + 1 u k + 1 āˆ‚ x Ļ• ) d x d t + āˆ« R u c ( x , 0 ) Ļ• ( x , 0 ) d x = āˆ’ āˆ« 0 + āˆž āˆ« R Ļ• [ āˆ‚ t u c + u c k āˆ‚ x u c ] d x d t = āˆ’ āˆ« 0 + āˆž āˆ« R Ļ• sign ( x āˆ’ c t ) [ c u c āˆ’ u c k + 1 ] d x d t .
(4.2)

Since u c (x,t)= c 1 / k e āˆ’ | x āˆ’ c t | , we have that when x>ct,

sign(xāˆ’ct) [ c u c āˆ’ u c k + 1 ] = c 1 + 1 k [ e c t āˆ’ x āˆ’ e ( k + 1 ) ( c t āˆ’ x ) ]
(4.3)

and when xā‰¤ct,

sign(xāˆ’ct) [ c u c āˆ’ u c k + 1 ] = c 1 + 1 k [ e ( x āˆ’ c t ) āˆ’ e ( k + 1 ) ( x āˆ’ c t ) ] .
(4.4)

By using ( 1 āˆ’ āˆ‚ x 2 ) āˆ’ 1 f=pāˆ—f and (4.1), we have that

āˆ« 0 + āˆž āˆ« R Ļ• āˆ‚ x ( 1 āˆ’ āˆ‚ x 2 ) āˆ’ 1 [ k 2 + 2 k k + 1 u c k + 1 ] d x d t = āˆ« 0 + āˆž āˆ« R Ļ• āˆ‚ x p āˆ— [ k 2 + 2 k k + 1 u c k + 1 ] d x d t = āˆ’ k 2 + 2 k 2 ( k + 1 ) c 1 + 1 k āˆ« 0 + āˆž āˆ« R x āˆ« R y Ļ• sign ( x āˆ’ y ) e āˆ’ | x āˆ’ y | e āˆ’ ( k + 1 ) | y āˆ’ c t | d y d x d t .
(4.5)

Now, we compute

āˆ’ k 2 + 2 k 2 ( k + 1 ) c 1 + 1 k āˆ« R y sign ( x āˆ’ y ) e āˆ’ | x āˆ’ y | e āˆ’ ( k + 1 ) | y āˆ’ c t | d y : = I 1 + I 2 + I 3 .
(4.6)

When x>ct, we have that

I 1 = āˆ’ k 2 + 2 k 2 ( k + 1 ) c 1 + 1 k āˆ« āˆ’ āˆž c t e āˆ’ ( x āˆ’ y ) e ( k + 1 ) ( y āˆ’ c t ) d y = āˆ’ k 2 + 2 k 2 ( k + 1 ) c 1 + 1 k e āˆ’ ( x + ( k + 1 ) c t ) āˆ« āˆ’ āˆž c t e ( k + 2 ) y d y = āˆ’ k 2 ( k + 1 ) c 1 + 1 k e c t āˆ’ x
(4.7)

and

I 2 = āˆ’ k 2 + 2 k 2 ( k + 1 ) c 1 + 1 k āˆ« c t x e āˆ’ ( x āˆ’ y ) e āˆ’ ( k + 1 ) ( y āˆ’ c t ) d y = āˆ’ k 2 + 2 k 2 ( k + 1 ) c 1 + 1 k e āˆ’ ( x āˆ’ ( k + 1 ) c t ) āˆ« c t x e āˆ’ k ( k + 2 ) k + 1 y d y = āˆ’ k + 2 2 ( k + 1 ) c 1 + 1 k [ e c t āˆ’ x āˆ’ e ( k + 1 ) ( c t āˆ’ x ) ]
(4.8)

and

I 3 = k 2 + 2 k 2 ( k + 1 ) c 1 + 1 k āˆ« x āˆž e ( x āˆ’ y ) e āˆ’ ( k + 1 ) ( y āˆ’ c t ) d y = k 2 + 2 k 2 ( k + 1 ) c 1 + 1 k e ( x + ( k + 1 ) c t ) āˆ« x āˆž e āˆ’ ( k + 2 ) y d y = k 2 ( k + 1 ) c 1 + 1 k e ( k + 1 ) ( c t āˆ’ x ) .
(4.9)

Thus, when x>ct, from (4.7)-(4.9), we have that

I 1 + I 2 + I 3 = āˆ’ k 2 ( k + 1 ) c 1 + 1 k e c t āˆ’ x āˆ’ k + 2 2 ( k + 1 ) c 1 + 1 k [ e c t āˆ’ x āˆ’ e ( k + 1 ) ( c t āˆ’ x ) ] + k 2 ( k + 1 ) c 1 + 1 k e ( k + 1 ) ( c t āˆ’ x ) = āˆ’ c 1 + 1 k [ e ( k + 1 ) ( c t āˆ’ x ) āˆ’ e c t āˆ’ x ] .
(4.10)

Similarly, when x<ct, we can obtain

I 1 + I 2 + I 3 = c 1 + 1 k [ e ( k + 1 ) ( x āˆ’ c t ) āˆ’ e x āˆ’ c t ] .
(4.11)

From (4.3), (4.4) and (4.10) as well as (4.11), we have that

āˆ« 0 + āˆž āˆ« R [ u c Ļ• t + 1 k + 1 u c k + 1 Ļ• x + p āˆ— [ k 2 + 2 k k + 1 u c k + 1 ] Ļ• x ] d x d t + āˆ« R u c ( x , 0 ) Ļ• ( x , 0 ) d x = 0 .
(4.12)

Thus, u c = c 1 / k e āˆ’ | x āˆ’ c t | is the solution in the sense of (4.12). For c>0, let u c (x,t)= c 1 / k e āˆ’ | x āˆ’ c t | . Thus u c (x,t) is the solitary wave for (1.1) (or (1.4)).

5 Proof of Theorem 1.3

Since when Q= k ( k + 2 ) k + 1 in (1.4), (1.4) possesses the peaked solitary wave c 1 / k e | x āˆ’ c t | , Theorem 1.3 can be proved similarly to Proposition 4 of [19].

6 Proof of Theorem 1.4

Since when Q= k ( k + 2 ) k + 1 in (1.4), (1.4) possesses the peaked solitary wave c 1 / k e | x āˆ’ c t | , 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 s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)).

Proof of Theorem 1.5 Applying Ī” q to (1.4) yields

( āˆ‚ t + u k āˆ‚ x ) Ī” q u= [ u k , Ī” q ] āˆ‚ x u+QP(D) Ī” q [ k 2 + 2 k k + 1 u k + 1 ] .
(7.1)

From (2.54) of page 112 in [24], since s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)), we have that

āˆ„ 2 s q āˆ„ [ u k , Ī” q ] āˆ‚ x u āˆ„ L p āˆ„ ā„“ r ā‰¤C āˆ„ u x āˆ„ L āˆž k āˆ„ u āˆ„ B p , r s ā‰¤C āˆ„ u āˆ„ B p , r s k + 1 .
(7.2)

By using (4) of Lemma 2.2 and P(D) is an S āˆ’ 1 -multiplier, since s>1+ 1 p (or sā‰„1+ 1 p if r=1 with pāˆˆ[1,+āˆž)), we have that

āˆ„ P ( D ) [ k 2 + 2 k k + 1 u k + 1 ] āˆ„ B p , r s ā‰¤C āˆ„ u x āˆ„ L āˆž k āˆ„ u āˆ„ B p , r s ā‰¤C āˆ„ u āˆ„ B p , r s k + 1 .
(7.3)

Going along the lines of the proof of Proposition A.1 of [18], from (7.2) and (7.3), we have that

āˆ„ u āˆ„ B p , r s ā‰¤ āˆ„ u 0 āˆ„ B p , r s +C āˆ« 0 t āˆ„ u x āˆ„ L āˆž k āˆ„ u āˆ„ B p , r s dĻ„
(7.4)
ā‰¤ āˆ„ u 0 āˆ„ B p , r s +C āˆ« 0 t āˆ„ u āˆ„ B p , r s k + 1 dĻ„.
(7.5)

Solving (7.4) yields

āˆ„ u āˆ„ B p , r s ā‰¤ e c āˆ« 0 t āˆ„ u x āˆ„ L āˆž k d Ļ„ āˆ„ u 0 āˆ„ B p , r s .
(7.6)

Solving (7.5) yields

āˆ„ u āˆ„ B p , r s ā‰¤ āˆ„ u 0 āˆ„ B p , r s ( 1 āˆ’ C k t āˆ„ u 0 āˆ„ B p , r s k ) 1 / k .
(7.7)

Assume that T ā‹† is the maximal time of existence of the solution to problem (1.4)-(1.5). If T ā‹† <āˆž, we claim that

āˆ« 0 T ā‹† āˆ„ u x āˆ„ L āˆž k dĻ„=+āˆž.
(7.8)

We prove the claim (7.8) by contradiction. If (7.8) is untrue, then from (7.8), we have that

āˆ„ u ( T ā‹† ) āˆ„ B p , r s <āˆž,
(7.9)

which contradicts the fact that T ā‹† 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 T ā‹† ā‰„ 1 C k āˆ„ u 0 āˆ„ B p , r s k . Moreover, (7.7) ensures the validity of (3.8).

The proof of Theorem 1.5 is completed.ā€ƒā–”

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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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