ON THE INITIAL VALUE PROBLEM ASSOCIATED TO A GENERALIZATION OF THE REGULARIZED BENJAMIN-ONO EQUATIO

International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN (P): 2249-6955; ISSN (E): 2249-8060 Vol. 8, Issue 4, Aug 2018, 7-20 © TJPRC Pvt. Ltd.

ON THE INITIAL VALUE PROBLEM ASSOCIATED TO A GENERALIZATION OF THE REGULARIZED BENJAMIN-ONO EQUATION JOHN BOLANOS & GUILLERMO RODRÍGUEZ-BLANCO Department of Mathematics, National University of Colombia, Bogota, Colombia ABSTRACT Our aim is to establish local and global well-posedness results in weighted Sobolev spaces

,

(ℝ) via

contraction principle and prove a unique continuation property for a generalization of the regularized Benjamin-Ono Equation.

MATH SUBJECT CLASSIFICATION: 35A01, 35A02, 35B60, 35B65, 35G25 & 42B35 KEYWORDS: Local and Global Well-Posedness, Unique Continuation Principle & Regularized Benjamin-Ono Equation

1. INTRODUCTION Many phenomena that occur in Physics and Engineering are modeled by partial differential equations.

Some kind of them, some very remarkable, such like the KdV equation, B-O equation, Schrödinger equation, to mention some of them, are the partial differential equations of evolution, whose name is due to the fact that one of

Original Article

Received: Jun 05, 2018; Accepted: Jun 25, 2018; Published: Jul 17, 2018; Paper Id.: IJMCARAUG20182

the independent variables is time. These models are also important in Mathematics, since they lead to problems such as, local and global well-posedness, stability of solitary waves, principles of single continuation, propagation of regularity, to mention some of them; whose solutions rescue classic techniques of Analysis, as they also give rise to new ideas that have led to their solution. In this work, we will deal with the initial value problem associated with a generalization of the regularized Benjamin-Ono equation (gr-BO). More accurately, We consider the problem:

+ + + = 0, (0, ) = ( ) ∈ (ℝ),

When

∈ ℝ, 0 <

< 1,

(1)

= (| | ! )∨ , is the homogeneous derivative of order # ∈ ℝ; .̂ is the Fourier transform and ∨ its

where, inverse.

,

= 0, equation (1) is the well-known regularized Benjamin-Ono equation (r-BO). Results for this

equation about well-posedness in Sobolev spaces

(ℝ), for

# > 1/2 and weighted Sobolev spaces ℱ ,* (ℝ) for

# > 1/2, 0 ≤ , < 5/2 together with unique continuation principles were obtained by Germán Fonseca, Guillermo

Rodríguez-Blanco and Wilson Sandoval in [5]. Following the ideas of this work, we obtained similar results for (1), and in order to make the reading of this work more enjoyable, we present a series of preliminaries in the following section.

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John Bolaños & Guillermo Rodríguez-Blanco

2. PRELIMINARIES

Theorem 1: If 0 < . < 1 and 1 < / < ∞, then ‖

2

( 3)‖45 ≤ 6(‖3‖7 ‖

‖4 5 + ‖ ‖7 ‖

2

Proof. See [7].

Definition 2: Let . ∈ (0,1) and

9 2 ( ) = :;ℝ<

|=( )>=(?)|@ | >?|<A@B

CDE

2

3‖45 )

(2)

measurable on ℝ8 with complex values. The Stein derivative is defined by:

/F

(3)

Definition 3: For # ∈ ℝ. We note by G (ℝ8 ) the space of all functions H

GH (ℝ8 ). The norm in this space is given by ‖ ‖

,H

= K(1 − Δ)@ K M

for each

on GH (ℝ8 )

∈ GH (ℝ8 ) such that (1 − J)

/F

∈

45

The following theorems characterize the spaces G (ℝ8 ) in terms of the Stein derivative: H

Theorem 4: Let . ∈ (0,1) and 2/N + 2. ≤ / < ∞. Then, ∈ GH (ℝ8 ).

1.

∈ G2 (ℝ8 ) if and only if H

2. 9 2 ( ) ∈ GH (ℝ8 ). with, ‖ ‖2,H = O(1 − Δ)@ O B

45

≅ ‖ ‖4 5 + ‖

2

‖45 ≅ ‖ ‖45 + ‖9 2 ‖45

Proof. See [9] or [10]

Theorem 5: Let . ∈ (0,1) and 1 ≤ / < ∞. If , 3: ℝ8 → ℂ are measurable functions, then ‖9 2 ( 3)‖4@ ≤ ‖ 9 2 3‖4@ + ‖39 2 ‖4@

(4)

Proof. See Proposition 1 in [8] Proposition 6: For . ∈ (0,1) ‖9 2 ‖7 ≤ T2 (‖ ‖7 + ‖∂

‖7 )

(5)

Proof. It is a direct consequence of the definition of the Stein derivative (3)

Proposition 7: Let V( ) and W( ) polynomials of degree X and N respectively with 0 < X < N, for . ∈ (0,1),

∈ GF (ℝ) and K

2

Y

Z

∈ 6(ℝ) then

: ⋅ EK ≤ T(‖ ‖\ + ‖9 2 ‖\ ) Y

Z

\

Proof. See Proposition 2.11 in [3]

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On the Initial Value Problem Associated to a Generalization of the Regularized Benjamin-Ono Equation

9

The following lemmas are important to obtain results for (1) on Sobolev spaces with fractional weights, in which the Stein derivative plays a leading role.

Lemma 8: Let . ∈ (0,1), ^_`a

> 0. Then,

9 2 (] bA|a|bAc ) ≤ 6( , .) 2 (6) for all > 0.

Proof. See proposition 2.13 in [3] Proposition 9 Let d ∈ 6\7, a function such taht supp d ⊆ [−2,2] and d ≡ 1 in

(−1,1). For any . ∈ (0,1) and i > 0,

|k| ≪ 1, T|k|j>2 + T , i ≠ ., p b nT(−ln|k|)@ , |k| ≪ 1, i = ., 9 2 :| |j d( )E (k)~ u |k| ≫ 1, o bAB , n|v|@ m

with 9 2 (| |j d( ))(⋅) continous in k ∈ ℝ − {0}. In particular, one has that 9 2 ( |j d( )) ∈ GF (ℝ) if and only if . < i + 1/2.

(7)

Similar result holds for 9 2 (| |j sgn( )d( )).

Proof. See proposition 2.9 in [4].

Proposition 10 If !(0) = 0 and 0 < ‚ < 1,

ƒ

… „ †sgn(

) !‡ƒ ≤ ƒ !ƒ + ƒ \ \

„

!ƒ

\

Proof. See proposition 2.19 in [3]

3. LOCAL WELL-POSEDNESS ON ˆ (ℝ) For convenience we will write the initial value problem (1) in the following form: =‰ + ( ) (0) = ∈ (ℝ),

(8)

which is equivalent to the Integral Equation, ( ) = Š( ) + ;\ Š( − ‹) ( (‹))C‹, where,

‰ = − ∂ (1 + Š( ) = ] Œ

)> ,

^_Ž

( ) = ‰(

= •] bA|Ž|bAc ••

∨

F

)

Proposition 11: The operator ‰ = −‘ (1 + Proof. Let

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∈

(ℝF )

(9)

)> is bounded on

(ℝ), if

≥ 0.

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John Bolaños & Guillermo Rodríguez-Blanco

‖‰( )‖F = ‖− ∂ (1 + = ;“@ (1 +

≤ ;“@ (1 +

F

F

) ”

)>

•„

|„|bAc

F

‖F

•” C

) | • |F C = ‖ ‖F

if

≥ 0 we have

Proposition 12: The application Š: ℝ ↦ ›(

(ℝ)) defined by Š( ) = ] Œ

for

∈

(ℝ) and ∈ [0, ∞) is a

strongly continuous unitary group.

Proof. The proof is simple and direct, for that reason we do not present it here.

Proposition 13: If # > 1/2 and

≥ 0, the fucntion ( ) = ‰(

‖ ( ) − (œ)‖ ≤ G(‖ ‖ , ‖œ‖ )‖ − œ‖ ,

for all , œ ∈

F

) satisfies the condition of local lipschitz, i.e,

(ℝ), where G(‖ ‖ , ‖œ‖ ) = ‖ ‖ + ‖œ‖

Proof. Since

(10)

(ℝ) is a Banach algebra for # > 1/2, the proof is a consequence of this fact and the

proposition (11).

To proof the existence of a solution of (8) we consider the set:

• (ž, Ÿ, ) = { ∈ 6([0, ž];

(ℝ)); ‖ ( ) − Š( ) ‖ ≤ Ÿ},

which is a complete metric subspace of 6([0, ž];

C (œ, ¡) = sup ∥ œ( ) − ¡( ) ∥ = ∥ œ − ¡ ∥ ∈[\,¤]

) with the metric

(11)

,7

and the application Ψ(œ)( ) = Š( ) + ;\ Š( − ‹) †œ(‹)‡C‹ Proposition 14: If

≥ 0 and # > 1/2, then the application § satisfy:

1. Exists ž (‖ ‖ , Ÿ) ≥ 0 such that Ψ(œ)( ) ∈ • (ž , Ÿ, )

2. Exists žF (‖ ‖ , Ÿ) ≥ 0 such that Ψ(œ)( ) is a contraction.

Proof. Part 1: ‖Ψ(œ)( ) − Š( ) ‖ ≤ ;\ ƒŠ( − ‹) †œ(‹)‡ƒ C‹ ≤ 6 ;\ ƒ†œ(‹)‡ƒ C‹ F

≤ 6 (Ÿ + ‖ ‖ )F

choosing ž1 > 0 such that the right side of 12 is less than Ÿ we get the result.

(12)

Part 2: ‖Ψ(œ)( ) − Ψ(¡)( )‖ ≤ ;\ KŠ( − ‹) : †œ(‹)‡ − †¡(‹)‡EK C‹

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On the Initial Value Problem Associated to a Generalization of the Regularized Benjamin-Ono Equation

11

≤ ;\ K: †œ(‹)‡ − †¡(‹)‡EK C‹by(10)

≤ 6 ;\ (‖œ(‹)‖ + ‖¡(‹)‖ )‖œ(‹) − ¡(‹)‖ C‹

≤ 26 (Ÿ + ‖ ‖ ) ;\ ‖œ(‹) − ¡(‹)‖ C‹

≤ 6 (Ÿ + ‖ ‖ )‖œ − ¡‖

,7

Choosing žF > 0 such that 6 (Ÿ + ‖ ‖ )žF < 1 we get that Ψ is a contraction.

The above proposition together with Banach’s fixed point theorem implies the following theorem. ∈

Theorem 15: If

(ℝF ), # > 1/2 and

≥ 0, exists ž = ž(‖ ‖ , Ÿ) > 0 and

∈ 6([0, ž];

(ℝF )) that

satisfies the integral equation (9).

Uniqueness and continuous dependence are followed by standard methods. ‖ ( )‖bAc ~‖ ‖bAc . @

for # > 1/2 and let

∈

Lemma 16 Suppose that

∈ 6([0, ž],

(ℝ)) be the solution of (1), then

@

Proof. The equation (1) implies that (1 +

)

= −∂

− ∂ F

F

.

(13)

Now ‖ ‖FbAc ~ 〈 ª@ , ª@ 〉, b

@

b

where, ª = (1 + ¬

¬

〈ª@ , ª@ 〉 = 2〈(1 + b

b

= 2〈 ,−∂ = −2〈 , ∂

=0

− ∂ F

〉 −〈 ,∂

) . Therefore, it easily follows that )@ , (1 +

)@ ((1 +

b

F〉

)> (− ∂

b

− ∂ F

F

))〉

F〉

This implies the result.

Next theorem shows that the I.V.P (1) is globally well-posed in priori estimates of the Sobolev norm in

(ℝ), # > 1/2. The key point is to obtain a

(ℝ), # > 1/2 with the help of the a priori bound of the

bAc @

norm in Lemma

2.2 and the Kato-Ponce commutator, [6], | 3|

,H

≤ T(‖ ‖4- |3|

,H

+ ‖3‖4- | | ,H ) for # > 0, 1 < / < ∞,

and the Brezis-Gallouet’s inequality, [2], which in dimension one is ‖ ‖4- ≤ T ¯1 + °log(1 + ‖ ‖ )‖ ‖b ± , for # > . @

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John Bolaños & Guillermo Rodríguez-Blanco

(ℝ), # > 1/2.

Theorem 17 The Cauchy problem (1) is g.w.p. in

((²)), the integral equation 9 implies that

∈

Proof. Let

‖ ( )‖ ≤ ‖ ‖ + ;\ ‖ (‹)‖7 ‖ (‹)‖ C‹

≤ ‖ ‖ + T\ ;\ †1 + °log(1 + ‖ (‹)‖ ) ‡‖ (‹)‖ C‹ =: Ψ( ),

where T\ only depends on ‖ ‖bAc . The above lemma and this inequality imply that @

Ψ ³ ( ) = T\ †1 + °log(1 + ‖ (‹)‖ ) ‡‖ (‹)‖ ≤ T\ •1 + ´log†1 + Ψ( )‡ • Ψ( ) ≤ T\ :1 + log†1 + Ψ( )‡E Ψ( )

Then, there exists T > 0 such that, ¬

¬

log :1 + log†1 + Ψ( )‡E ≤ T

and hence, there are constants TF > 0 and Tµ > 0 such that for every ∈ [0, ž], ‖ ( )‖ ≤ ] u@¶

4. THE PROBLEM GR-BO ON WEIGHTED SPACES

·¸ `

.

,

In this section, we study the IVP (1), on weighted spaces ℱ ,* , in which we establish local well-posedness and

single continuation principles. To study the local well-posedness on these spaces we need to bound the operator ‰ = − ∂ (1 +

)> and the group Š( ) en ℱ ,* .

Keeping this in mind, we calculate the first derivatives of the function ∂„ : ∂F„ :

∂µ„ ¯

>•„

|„|bAc >•„

|„|bAc

−¼ 1+| |

E= E=

>•( > |„|bAc ) (

>• (

±=

|„|bAc )@ (

+

)|„|@cAb ¹º»(„) |„|bAc )¸

•(

)(F

¼ (1 + )(2 + )| | (1 + | | )½

µ

(

Proposition 18: The operator ‰ = −‘ (1 +

>•„

|„|bAc

.

)|„|c ¹º»(„)

|„|bAc )¸

−

¼(1 + )(4 F + 8 + 6)| |F (1 + | | )½

+

¼ (1 + )(2 + )| | (1 + | | )½

)> is bounded ℱ ,* (ℝ) for 0 ≤ , < 5/2 +

>

< 3.

Proof.

‖‰ ‖ℱM, = ‖‰ ‖ + ‖‰ ‖4@Â

The first term is bounded in a similar way to the proposition 11. For the second term lets suppose that , = 2,

‖‰ ‖4@@ = K∂F„ : ≤K

(

(

>•„

|„|bAc

•EK

)|„|@cAb ¹º»(„)• |„|bAc )¸

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\

•K + K \

(

)(F (

)|„|c ¹º»(„)•

|„|bAc )¸

•K +2 K \

(> (

|„|bAc )•

|„|bAc )@

∂„ •K + K: \

>•„

|„|bAc

E ∂F„ •K

\

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13

≤ T :‖ •‖\ + ƒ∂„ •ƒ + ƒ∂F„ •ƒ E ≤ T‖ ‖4@@,à \

\

Stein-weiss interpolation theorem (see theorem 5.4.1 in [1]) together with the previous inequality allows us to

conclude ‖

*

‰ ‖\ ≤ ‖

‖\ for 0 ≤ , ≤ 2.

*

For the case 2 < , < + , let , = 2 + . with 0 < . < 1 and . < 1/2 + .

‖‰ ‖4@Â

Ä

=‖

b ,Ã

=Å

2 „

≤ TK

*b

•∂F„ 2 „

:

F

‰ ‖\

:

• E•Å

>•„

|„|bAc

\

•EK + T K

|„|@cAb ¹º»(„) (

F

|„|bAc )¸

2 „

\

= ‰ + ‰F + ‰µ + ‰½

:

•EK +T K

|„|c ¹º»(„)

(

|„|bAc )¸

\

2 „

:

> (

|„|bAc

|„|bAc )@

∂„ • EK + K \

2 „

:

„

|„|bAc

∂F„ •EK

\

To bound ‰F we use the function d defined in proposition 9

‰F = K

2 „

:

(

|„|bAc )¸

(

|„|bAc )¸

≤K

2 „

:

≤K

2 „

:

|„|c ¹º»(„)

(

|„|bAc )¸

\

\

E ⋅ •K + K \

≤ ‰F, + ‰F,F + ‰F,µ

Let 3( ) =

|„|c ¹º»(„)Æ(„) (

(14)

•EK + K

|„|c ¹º»(„)Æ

|„|c ¹º»(„)Æ

•EK

|„|bAc )¸

2 „

:

|„|c ¹º»(„)( >Æ) (

|„|bAc )¸

|„|c ¹º»(„)Æ

|„|bAc )¸

(

2 „

• EK

\

•K + K \

2 „

:

|„|c ¹º»(„)( >Æ) (

|„|bAc )¸

•EK

\

, as a consequence of the proposition 7 and proposition 9 we get

important to note that to use proposition 9, we must impose the condition that . < 1/2 + . ƒ

2 „ (3(

))ƒ = K

2 „

\

:

|„|c ¹º»(„)Æ(„) (

|„|bAc )¸

2 „ 3(

) ∈ GF (ℝ). It is

EK

\

≤ T :‖| | sgn( )d( )‖\ + ƒ9„2 (| | sgn( )d( ))ƒ E

≤ Ç if . <

To bound ‰F, (‰F, )F = K =ƒ

2 „

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(

|„|bAc )¸

)) •ƒ

2 „ (3(

≤ Ç F ‖ •‖F4b /F

+ 1/2.

:;ℝ |(1 +

F

F

E •K

|„|c ¹º»(„)Æ(„)

2 „ (3(

≤ ‖ •‖F7 ƒ

≤Ç

:

\

\

(15)

\

))ƒ

(16)

F \

)

F */F

•( ) ⋅

(

@ )Â/@

|C E

F

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John Bolaños & Guillermo Rodríguez-Blanco

≤Ç

/F

¯†;ℝ |(1 +

)

F */F

≤ T‖ ‖F\ + T‖ ‖F4@Â .

• ( )|F C ‡

/F

:;ℝ

@ )Â

(

/F F

C E

±

The bound for the term ‰F,F is immediate and the bound for the term ‰F,µ is easier, because we are on the set

where the function (1 − d( )) ≠ 0, so ‰F,µ = K

2 „

:

|„|c ¹º»(„)( >Æ(„)) (

|„|bAc )¸

≤K

2 „

:

|„|c ¹º»(„)( >Æ(„))

≤K

2 „

:

|„|c ¹º»(„)( >Æ(„))

(

|„|bAc )¸

(

|„|bAc )¸

≤ T‖ ‖\ + T‖ ‖4@Â

•EK

\

E •K + K

|„|c ¹º»(„)( >Æ(„))

\

(

EK ‖ • ‖\ + K 7

|„|bAc )¸

2 „

•K

|„|c ¹º»(„)( >Æ(„)) (

|„|bAc )¸

\

K ƒ

2 „

7

•ƒ

\

b ,Ã

The bound for ‰ is obtained in a similar way as the bound for ‰F , with the difference that the bound condition is

. < (2 + 1) + 1/2, which is true for all 0 <

< 1. The bounds for the terms ‰µ and ‰½ are a direct consequence of the

proposition 7. ^_Ž

∨

Now we need to bound the group Š( ) = •] bA|Ž|bAc •• = (È( , ) •)∨ on weighted spaces ℱ * .

For this purpose it is necessary to calculate the partial derivatives of È( , ):

∂„ È( , ) =

>•( > |„|bAc )

∂F„ È( , ) = :

∂µ„ È( , ) = :

+

(

)(F

|„|bAc )@

(

>• ( • (

( (

È

)|„|@cAb ¹º»(„) |„|bAc )¸

)(F

)|„|bA¸c

|„|bAc )É

)( > |„|bAc )|„|c ¹º»(„) @ (

|„|bAc )Í

+

−

−

•(

•( F (

)(F (

)|„|c ¹º»(„)

|„|bAc )¸

)(½ @ Ê (

Ë)|„|@c

|„|bAc )É

+

+

( > |„|bAc )@ @ (

• (

|„|bAc )É

(

)( > |„|bAc )|„|c ¹º»(„) @ (

|„|bAc )É

)(F

−

)|„|c^b

|„|bAc )É ½(

The following proposition bounds the group Š( ) for integer index. ^_Ž

EÈ

−

(

)( > |„|bAc )|„|@cAb Ì8(„) @ (

|„|bAc )Í

)( > |„|bAc )@ |„|c ¹º»(„) @ (

|„|bAc )Í

+

•( > |„|bAc )¸ ¸ (

|„|bAc )Î

EÈ

∨

Proposition 19: Let Š( ) = •] bA|Ž|bAc •• . If , = 1 or , = 2 then ||Š( ) ||ℱM, ≤ V* ( )|| ||ℱM,Â

where V* ( ) is a polynomial of degree ,.

Proof. We show the case , = 2. The case , = 1 is similar.

‖Š( ) ‖ℱM,@ = ‖Š( ) ‖ + ‖Š( ) ‖4@@ ≤‖ ‖ +‖

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F

Š( ) )‖\

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On the Initial Value Problem Associated to a Generalization of the Regularized Benjamin-Ono Equation

≤ ‖ ‖ + ƒ∂F„ (È •)ƒ

≤‖ ‖ +K

• (

+K

•(

+K

)(F (

(

È •K

)|„|@cAb ¹º»(„) |„|bAc )¸

È •K + K

)|„|c ¹º»(„)

|„|bAc )¸

\

\

( > |„|bAc )@ @

È ∂„ •K + ƒ∂F„ •ƒ

•( > |„|bAc ) (

\

|„|bAc )@

≤ T :‖ ‖ + ‖ •‖\ + ≤ V* ( )‖ ‖ℱM,@

F‖

15

\

(

|„|bAc )É

È •K

\

\

• ‖\ + ƒ∂„ •ƒ + ƒ∂F„ •ƒ E \

\

The following proposition bounds the group Š( ) for non-integer index. We use the Stein derivative defined in 3.

Proposition 20: Let Š( ) = •] Then,

∥ Š( ) ∥ℱM, ≤ 6( ) ∥

∨

^_Ž bA|Ž|bAc

•• , 0 ≤ , < 5/2 +

< 3.

∥ℱM, ,

where 6( ) is a continuous increasing function in t.

Proof.

∥ Š( ) ∥ℱM, =∥ Š( ) ∥ +∥ Š( ) ∥4@ ≤∥

∥ +∥ | |* Š( ) ∥\

Let’s suppose , = . with 0 < . < 1, using the properties of Stein derivative and lemma 8, we get:

∥ Š( ) ∥ℱM, =∥

∥ +∥ | |2 Š( ) ∥\

= ‖ ‖ + ƒ9„2 (È •)ƒ

\

≤ ‖ ‖ + ƒ9„2 È ⋅ •ƒ + ƒÈ ⋅ 9„2 •ƒ +‖T( , .)

≤ 6( ) ∥

2

\

•‖\ + ƒ9„2 •ƒ

∥ℱM,Â

\

\

now, let , = 1 + . with 0 < . < 1, using the proposition 7 we have

∥ Š( ) ∥ℱM, =∥

=‖ ‖ +ƒ ≤‖ ‖ +ƒ

≤‖ ‖ +K www.tjprc.org

∥ +∥ | | 2 „

∂„ (È •)ƒ

2 „ (∂„ È 2 „

2

:

>•( (

Š( ) ∥\

\

⋅ • )ƒ + ƒ \

|„|bAc )

|„|bAc )@

2 „ (È

⋅ ∂„ • )ƒ

\

È ⋅ •EK + ƒ9„2 È ⋅ ∂„ •ƒ +‖È‖7 ƒ9„2 (∂„ •)ƒ \

\

\

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16

John Bolaños & Guillermo Rodríguez-Blanco

≤ ‖ ‖ + T :‖È •‖\ + ƒ9„2 (È •)ƒ E + ƒT(.)

2

\

∂„ • ƒ

+ ƒ9„2 (∂„ • )ƒ

\

\

(17)

≤ T( ) :‖ ‖ + ƒÈ9„2 •ƒ + ƒ9„2 È •ƒ + ƒ∂„ •ƒ +ƒ∂„ • ƒ + ƒ9„2 ∂„ •ƒ E \

\

≤ T( )(‖ ‖ + ‖| |2 ‖\ + ‖| | ‖\ + ‖| | ≤ T( ) ∥

2

∥ℱM,Â

\

\

‖\ )

\

Now let’s suppose that , = 2 + ., with 0 < . < 1 and . < 1/2 +

∥ Š( ) ∥ℱM, =∥ ≤‖ ‖ +ƒ ≤‖ ‖ +ƒ

2 „

≤‖ ‖ +K +K

2 „

+ƒ

:

∥ +∥ | |F 2 Š( ) ∥\

∂F„ ( •)ƒ

2 F „ (∂„ È

⋅ •)ƒ + ƒ

2 „

|„|bAc )¸

:

\

|„|@cAb ¹º»(„) (

( > |„|bAc )@ @

2 „ (È

(

\

|„|bAc )É

⋅ ∂F„ •)ƒ

2 „ (∂„ È

⋅ ∂„ •)ƒ + ƒ

È • EK + K \

È •EK + K

2 „

\

:

2 „

:

\

|„|c ¹º»(„)

(

|„|bAc )¸

( > |„|bAc ) (

2 „ (È

|„|bAc )@

⋅ ∂F„ •)ƒ

È •EK

\

\

È ∂„ •EK

\

\

≤ ‖ ‖ + Ï + ÏF + ϵ + Ͻ + ÏÄ

(18)

First lets bound ÏF , using the function d( ) defined in proposition 9: ÏF = K

2 „

:

≤K

:

|„|c ¹º»(„)Æ(„)

È •EK + K

2 „

:

E •K + ƒ

2 „

≤ K + K

2 „

È •EK

|„|c ¹º»(„)

(

|„|bAc )¸

(

|„|bAc )¸

|„|c ¹º»(„)Æ(„)

:

(

|„|bAc )¸

\

|„|c ¹º»(„)( >Æ(„)) (

|„|bAc )¸

≤ ÏF, + ÏF,F + ÏF,µ .

\

\

È •EK

2 „

:

2 „ (È

|„|c ¹º»(„)( >Æ(„)) (

• )ƒ

|„|bAc )¸

• EK

\

\

\

The term ÏF, was bounded on the equation 15 and the term ÏF,F was bounded on the equation 17, to bound ÏF,µ

we have in mind that we are on the set where the function 1 − d( ) ≠ 0, i.e, ÏF,µ = K9„2 : ≤K

|„|c ¹º»(„)( >Æ(„)) (

|„|bAc )¸

(

≤ TÏF,F + T‖ •‖\ .

\

(19)

K ƒ9„2 (È • )ƒ + K9„2 :

|„|c ¹º»(„)( >Æ(„)) |„|bAc )¸

È •EK

7

\

|„|c ¹º»(„)( >Æ(„)) (

|„|bAc )¸

E È •K

\

Ï is bounded in a similar way as ÏF but with the condition . < (2 + 1) + 1/2, which is true for 0 <

The bounds for the terms ϵ , Ͻ and ÏÄ are obtained from the lemma 8 and the proposition 7.

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On the Initial Value Problem Associated to a Generalization of the Regularized Benjamin-Ono Equation

17

< 3, then exists ž = ž(‖ ‖ℱM, , Ÿ) > 0 and an

∈ ℱ ,* (ℝ) with # > 1/2 and 0 ≤ , < 5/2 +

Theorem 21: If

∈ 6([0, ž]; ℱ ,* ) which satisfy the integral equation 9, also, the function

unique function

→

associated to the

equation 1 is continuous.

Proof. By propositions 18 and 20, where the operator ‰ and the group Š( ) were bounded in ℱ ,* (ℝ), and (ℝ) we get the local well-posedness.

following the ideas of the theory on 4.1. Unique continuation of solutions

∈ ℱ ,* (ℝ) with # > 1/2 and 0 ≤ , ≤ 5/2 +

Theorem 22: If

∈ 6([0, ž]; ℱ ,* ) the solution of the initial value problem 1, with (0) = , such that

from theorem 21 are satisfied, let ;ℝ

( )C ≥ 0, if for two times =

F

F

=0<

F

b @ < ž we have ( • ) ∈ ℱÄ/F

,

∈ 6([0, ž]; ℱ ,* )) the solution of 1. multiplying

Proof. Let

Š( ) +

F

< 3, where the conditions for well-posedness

;\ Š( − ‹)‰( „

We will analyze the derivative

/F

F

(‹))C‹.

for 0 ≤

F

,\

Ð = 1,2 then

≡ 0.

by the integral equation 9 we have,

< 1/2 on both sides of equality. Applying the Fourier transform

to the first term we get:

∂F„ (È •) = ∂F„ È ⋅ • + 2 ∂„ È ⋅ ∂„ • + È ⋅ ∂F„ • =

• (

(

)|„|@cAb ¹º»(„) |„|bAc )¸

È•+

•(

)(F (

= 6 + 6F + 6µ + 6½ + 6Ä

)|„|c ¹º»(„)

|„|bAc )¸

È• +

( > |„|bAc )@ @ (

|„|bAc )É

From the bounds for the terms Ï , ϵ , Ͻ and ÏÄ in (18), we have

the term 6F we use the function d from proposition 9:

=

6F = −

•(

)(F (

Ñc |„|c ¹º»(„)Æ(„) (

|„|bAc )¸

È•

)|„|c ¹º»(„)

|„|bAc )¸

È•+

Ñc |„|c ¹º»(„)( >Æ(„)) |„|bAc )¸

(

È•−

„

/F

F•( > |„|bAc ) (

|„|bAc )@

È ∂„ • + È ∂F„ •

6• ∈ GF (ℝ) for ¼ = 1,3,4,5. To bound

È•

= 6F, + 6F,F whereÇ = −¼(1 + )(2 + ) Note that 6F, = Ç

„

/F

6F,F ∈ GF (ℝ) due to (19). To bound the term 6F, , we rewrite it in the following way:

|„|c ¹º»(„)Æ(„) (

|„|bAc )¸

È•

= Ç | | sgn( )d( ) • :

=

+

Since (

F

F.

„

/F

(

∈ GF (ℝ) and

Š( ) )∧ = 6 +

+

Ò

|„|bAc )¸

F

− 1E + Ç | | sgn( )d( ) •

+ 6F,F + 6µ + 6½ + 6Ä

then, www.tjprc.org

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18

John Bolaños & Guillermo Rodríguez-Blanco „

(

/F

F

Š( ) )∧ −

F)

∈ GF (ℝ),

that is, „

((

/F

F

Š( ) )∧ − Ç | | sgn( )d( ) •( )) ∈ GF (ℝ).

F

For the term of the integral let ‖

F

Š( )‰(œ)‖\ = K∂F„ :È

∂F„ :È

+

>•„

|„|

ϥE =

•( > |„|bAc )@ „ @

+

|„|bAc )Í

(

(

|„|bAc )@

)|„|@cA@

(

(

/F

ϥEK , thus

Èœ• −

F( > |„|bAc )@ (

È ∂„ œ• +

|„|bAc )É >•„

|„|bAc

= Š + ŠF +⋅⋅⋅ +ŠÔ

„

= œ and the Plancherel theorem implies

|„|bAc

|„|bAc )É

Èœ• +

>F•( > |„|bAc )

>•„

(20)

\

(

(

)|„|bAc

)(F

Èœ• +

È ∂F„ œ•

|„|bAc )É

Èœ•

F( > |„|bAc )„ (

|„|bAc )¸

È ∂„ œ• +

>• (

(

)|„|@cAb ¹º»(„) |„|bAc )¸

Èœ• +

•(

)(F

(

)|„|c ¹º»(„)

|„|bAc )¸

Èœ•

Š• ∈ 6([0, ž]; GF (ℝ)), for ¼ = 2,3, . . . ,9 ¼ ≠ 7.

Now we rewrite the term Š× : Š× =

= −Ç

•(

)(F

(

)|„|c ¹º»(„)

|„|bAc )¸

|„|c ¹º»(„)( >Æ(„)) (

|„|bAc )¸

Èœ•

Èœ• − Ç

= Š×, − Ç | | sgn( )d( ) : = Š×, + Š×,F + Š×,µ .

Since, „

/F

„

†(

/F

F

Š×, ,

„

/F

(

|„|c ¹º»(„)Æ(„) (

Ò

|„|bAc )¸

|„|bAc )¸

Èœ•

− 1E œ• − Ç | | sgn( )d( )œ•

Š×,F ∈ 6([0, ž]; GF (ℝ)) we have,

Š( )‰(œ))∧ − Š×,µ ‡ ∈ 6([0, ž]; GF (ℝ)). (21)

Note that 20 and 21, imply „

/F

†(

F

( ))∧ −

F

− Š×,µ ‡ ∈ GF (ℝ)

for all ∈ [0, ž]. By hypothesis, exists „

/F

F

such that ( F ) ∈ ℱ

, /F

, then

¯| | sgn( )d( ) :;\ @ • ( ) + œ•(‹, )C‹E± ∈ GF (ℝ). (22)

For convenience, we will write, ℎÙ( ) = :;\ @ •( ) + œ•(‹, )C‹E, that transforms the previous identity into „

/F

(| | sgn( )d( )ℎÙ( )) ∈ GF (ℝ),

which is the same as,

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On the Initial Value Problem Associated to a Generalization of the Regularized Benjamin-Ono Equation „

/F

Since, „

/F

then, „

/F

†| | sgn( )d( )(ℎÙ( ) − ℎÙ(0)) + | | sgn( )d( )ℎÙ(0)‡ ∈ GF (ℝ) †| | sgn( )d( )(ℎÙ( ) − ℎÙ(0))‡ ∈ GF (ℝ), †| | sgn( )d( )ℎÙ(0)‡ ∈ GF (ℝ)

Theorem 9 states that „

/F

19

2 „ (|

| sgn( )d( )) ∈ GF (ℝ) if and only if . < 1/2 + , then

†| | sgn( )d( )ℎÙ(0)‡ ∉ GF (ℝ),

unless, ℎÙ(0) = 0. This observation in our case becomes

;\

@

• (0) + œ•(‹, 0)C‹ = 0,

that is, @ ;\ ;ℝ ( ( ) + œ(‹, ))C C‹ = 0.

Since œ =

F

, we have

@ ;\ ;ℝ ( ( ) +

F

(‹, ))C C‹ = 0

This equality and the hypothesis, ;ℝ

( )C ≥ 0 imply

F

= 0 and therefore, ≡ 0.

Note 23 The previous unique continuation principle was obtained under the hypothesis 0 ≤

≥ 1/2, we will have to bound ‘„µ Š( ), but one of its terms is

for 1/2 ≤

< 1, the persistence is obtained for 0 ≤ , < 3.

• (

(

)(F

)|„|c^b

|„|bAc )É

< 1/2, as if

È, which has a singularity at zero. Thus,

CONCLUSIONS •

Using Banach’s fixed-point theorem we proved well-posedness in Sobolev spaces, for and weighted Sobolev spaces, for and.

•

We proved global well-posedness in Sobolev spaces, for.

•

The unique continuation principle was obtained, that is, if the initial data with and, and is the solution for the initial value problem (1), with and, $ and there also exists two times such that for then the solution is identically zero.

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20

John Bolaños & Guillermo Rodríguez-Blanco

REFERENCES 1.

J. Berg, J. Löfström, Interpolation spaces: an introduction, Grundlehren der mathematischen Wissenchaften, Springer-Verlag (1976).

2.

H. Brezis, T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis, 4(4) (1980), 677-681.

3.

J. Bolaños, El problema de Cauchy asociado a una generalización de la ecuación ZK-BBM, PhD Thesis, Universidad Nacional de Colombia (2018).

4.

G. Fonseca, F. Linares, G. Ponce, The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, (2013), 763-790.

5.

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval, Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces, arXiv preprint arXiv:1304.6454, (2013).

6.

T. Kato, G. Ponce, Commutator estimates and the euler and navier-stokes equations, Communications on Pure and Applied Mathematics, 41(7) (1988), 891-907.

7.

C. E. Kening, G. Ponce, L. Vega, Well-posedness and Scattering Results for the Generalized Korteweg-de Vries Equation via the Contraction Principle, Communications on Pure and Applied Mathematics, 46(4) (1993), 527-620.

8.

J. Nahas, G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation, Communications in Partial Differential Equations, 34(10) (2009), 1208-1227.

9.

E. M. Stein, The characterization of functions arising as potentials, Bulletin of the American Mathematical Society, 67(1) (1961), 102-104.

10. E. M. Stein, Singular integrals and differentiability properties of functions volume 2, Princeton university press (1970).

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