Some properties of solutions to the weakly dissipative b -family equation

Nonlinear Analysis: Real World Applications 13 (2012) 158–167

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Some properties of solutions to the weakly dissipative b-family equation Mingxuan Zhu, Zaihong Jiang ∗ Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China

article

abstract

info

Article history: Received 22 March 2011 Accepted 19 July 2011

In this paper, we investigate the b-family equation with a weakly dissipative term. First, we present a new criterion on the blow-up phenomenon of the solution. Then the global existence and the persistence property of the solution are also established. Finally, we discuss the infinite propagation speed for this equation. © 2011 Elsevier Ltd. All rights reserved.

Keywords: b-family equation Blow-up Persistence property Infinite propagation speed

1. Introduction Recently, Holm and Staley [1] studied the exchange of stability in the dynamics of solitary wave solutions under changes in the nonlinear balance in a 1 + 1 evolutionary partial differential equation related both to shallow water waves and to turbulence. They derived the following equations (the b-family equation): yt + uyx + bux y = 0,

t > 0, x ∈ R,

(1.1)

where u(x, t ) denotes the velocity field and y(x, t ) = u − uxx . A detailed description of the corresponding strong solutions to (1.1) with the initial data u0 was given by Zhou in [2]. In this work, a sufficient condition in the profile of the initial data for blow-up in finite time is established. The necessary and sufficient condition for blow-up is still a challenging problem for us at present. More importantly, Theorem 3.1 in [2] means that no matter what the profile of the compactly supported initial datum u0 (x) is (no matter whether it is positive or negative), for any t > 0 in its lifespan, the solution u(x, t ) is positive at infinity and negative at negative infinity. It is a very good property for the b-family equation. The famous Camassa–Holm equation [3] and Degasperis–Procesi equation [4] are the special cases with b = 2 and b = 3, respectively. There have been extensive studies on the two equations, cf. [5–11]. In this paper, we consider the following weakly dissipative b-family equation, yt + uyx + bux y + λy = 0,

t > 0, x ∈ R,

(1.2)

where y = u − uxx , λ > 0 and λy = λ(u − uxx ) is the weakly dissipative term. 1

Let Λ = (1 − ∂x2 ) 2 , then the operator Λ−2 can be expressed by its associated Green’s function G =

Λ−2 f (x) = G ∗ f (x) =

∗

1 2

∫

e−|x−y| f (y)dy. R

Corresponding author. E-mail addresses: [email protected] (M. Zhu), [email protected], [email protected] (Z. Jiang).

1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.07.020

1 −|x| e 2

as (1.3)

M. Zhu, Z. Jiang / Nonlinear Analysis: Real World Applications 13 (2012) 158–167

159

So system (1.2) is equivalent to the following system ut + uux + ∂x G ∗



b 2

u2 +

3−b 2

u2x



+ λu = 0.

(1.4)

In [12], Niu and Zhang established the following local well-posedness result for (1.2) by using Kato’s theory. Theorem 1.1 ([12]). Given u0 ∈ H s (R), s >

3 ; 2

then, there exists a T and a unique solution u to (1.2) such that

u(x, t ) ∈ C ([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)). Then they present the precise blow-up scenario for solutions to (1.2). Theorem 1.2 ([12]). Assume that u0 ∈ H s (R), s > 2. If b =

1 , 2

then the solution of (1.2) will exist globally in time. If b >

then the solution blows up if and only if ux becomes unbounded from below in finite time. If b < time if and only if ux becomes unbounded from the above in finite time.

1 , 2

1 , 2

the solution blows up in finite

The paper is organized as follows. In Section 2, we establish a new criterion on blow-up for (1.2). A condition for global existence is found in Section 3. Persistence property is considered in Section 4. In Section 5, the infinite propagation speed will be established analogous to the b-family equation. It is worth pointing out that, recently, many works have been done for similar equations which have a dissipative term, cf. [13–17]. 2. Blow-up Motivated by Mckean’s deep observation for the Camassa–Holm equation [7], we can do the similar particle trajectory as



qt = u(q, t ), q(x, 0) = x,

0 < t < T , x ∈ R, x ∈ R,

(2.1)

where T is the life span of the solution, then q is a diffeomorphism of the line. Differentiating the first equation in (2.1) with respect to x, one has dqt dx

= qxt = ux (q, t ),

t ∈ (0, T ).

Hence, qx (x, t ) = exp

t

∫



ux (q, s)ds ,

qx (x, 0) = 1.

0

Since d dt

(y(q)qbx ) = [yt (q) + u(q, t )yx (q) − bux (q, t )y(q)]qbx = −λyqbx ,

It follows that y(q)qbx = y0 (x)e−λt .

(2.2)

In [12], a blow-up result was proved for odd initial data. Here, we establish sufficient conditions on more general initial data to guarantee blow-up for Eq. (1.2). Theorem 2.1. Let b ≥ 2. Suppose that u0 ∈ H 2 (R) and there exists a x0 ∈ R such that y0 (x0 ) = (1 − ∂x2 )u0 (x0 ) = 0, y0 ≥ 0(̸≡ 0) e −x 0

∫

x0

for x ∈ (−∞, x0 ) and

eξ y0 (ξ )dξ > 2λ and

−∞

ex0

∫

y0 ≤ 0(̸≡ 0)

for x ∈ (x0 , ∞),

(2.3)

∞

e−ξ y0 (ξ )dξ < −2λ.

(2.4)

x0

Then the corresponding solution u(x, t ) to Eq. (1.2) with u0 as the initial datum blows up in finite time. Proof. Suppose that the solution exists globally. Due to Eq. (2.2) and the initial condition (2.3), we have y(q(x0 , t ), t ) = 0, and



y(q(x, t ), t ) ≥ 0(̸≡ 0), y(q(x, t ), t ) ≤ 0(̸≡ 0),

for x ∈ (−∞, (q(x0 , t ), t )), for x ∈ ((q(x0 , t ), t ), ∞),

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M. Zhu, Z. Jiang / Nonlinear Analysis: Real World Applications 13 (2012) 158–167

for all t ≥ 0. Since u(x, t ) = G ∗ y(x, t ), one can write u(x, t ) and ux (x, t ) as ∞ 1 1 −x x ξ e y(ξ , t )dξ + ex e−ξ y(ξ , t )dξ , e 2 2 −∞ x ∫ ∫ 1 −x x ξ 1 x ∞ −ξ ux (x, t ) = − e e y(ξ , t )dξ + e e y(ξ , t )dξ . 2 2 −∞ x

∫

u(x, t ) =

∫

Consequently, u2x (x, t ) − u2 (x, t ) = −

x

∫

∞

∫

eξ y(ξ , t )dξ

−∞

e−ξ y(ξ , t )dξ , x

for all t > 0. For any fixed t, if x ≤ q(x0 , t ), then u2x (x, t ) − u2 (x, t ) = −

q(x0 ,t )

∫

eξ y(ξ , t )dξ −

−∞ ∞

× q(x0 ,t )

=

eξ y(ξ , t )dξ



x

∫

u2x

q(x0 ,t )

∫

e−ξ y(ξ , t )dξ +

q(x0 ,t )

∫

e−ξ y(ξ , t )dξ



x

(q(x0 , t ), t ) − u (q(x0 , t ), t ) − 2

∫

x

ξ

e y(ξ , t )dξ −∞

q(x0 ,t )

∫ +

ξ

e y(ξ , t )dξ

e−ξ y(ξ , t )dξ

x

∞

∫

q(x0 ,t )

x

q(x0 ,t )

∫

e−ξ y(ξ , t )dξ

≤ u2x (q(x0 , t ), t ) − u2 (q(x0 , t ), t ).

(2.5)

Similarly, for x ≥ q(x0 , t ), we also have u2x (x, t ) − u2 (x, t ) ≤ u2x (q(x0 , t ), t ) − u2 (q(x0 , t ), t ).

(2.6)

Combining (2.5) and (2.6) together, we get that for any fixed t, u2x (x, t ) − u2 (x, t ) ≤ u2x (q(x0 , t ), t ) − u2 (q(x0 , t ), t ),

(2.7)

for all x ∈ R. Differentiating Eq. (1.4) with respect to x, we obtain utx + uuxx −

b 2

2

u −

1−b 2

u2x



b

+G∗

2

2

u +

3−b 2

u2x



+ λux = 0.

Differentiating ux (q(x0 , t ), t ) with respect to t, where q is the diffeomorphism defined in (2.1)

∂t ux (q(x0 , t ), t ) = uxt (q(x0 , t ), t ) + uxx (q(x0 , t ), t )qt (q(x0 , t ), t )   b 1−b 2 b 2 3−b 2 = u2 (q(x0 , t ), t ) + ux (q(x0 , t ), t ) − λux (q(x0 , t ), t ) − G ∗ u (x, t ) + ux (x, t ) 2 2 2 2   b 2 1−b 2 b 2 3−b 2 = G∗ u (q(x0 , t ), t ) + ux (q(x0 , t ), t ) − u (x, t ) − ux (x, t ) − λux (q(x0 , t ), t ) 2 2 2 2   b−2 2 = G∗ (u (q(x0 , t ), t ) − u2x (q(x0 , t ), t ) − u2 (x, t ) + u2x (x, t )) 2   1 1 + G ∗ u2 (q(x0 , t ), t ) − u2x (q(x0 , t ), t ) − u2 (x, t ) − u2x (x, t ) − λux (q(x0 , t ), t ) 2

≤

1 2

u2 (q(x0 , t ), t ) −

1 2

2

u2x (q(x0 , t ), t ) − λux (q(x0 , t ), t ),

where we have used (2.7) and the inequality G ∗ (u2 (x, t ) + 21 u2x (x, t )) ≥ By (2.4), we find that

(u0x (x0 ) + λ) − (u0 (x0 ) + λ) = −e 2

2

−x 0

∫

x0

−∞

ξ

e y0 (ξ )dξ ×

 e

x0

∫

1 2 u . 2

∞

e x0

(2.8)

−ξ

y0 (ξ )dξ + 2λ



>0

M. Zhu, Z. Jiang / Nonlinear Analysis: Real World Applications 13 (2012) 158–167

161

and ∞

∫

(u0x (x0 ) + λ)2 − (u0 (x0 ) − λ)2 = −ex0

e−ξ y0 (ξ )dξ ×



x0

∫

e−x0

eξ y0 (ξ )dξ − 2λ



> 0.

−∞

x0

Claim. ux (q(x0 , t ), t ) < 0 is decreasing, (u(q(x0 , t ), t ) + λ)2 < (ux (q(x0 , t ), t ) + λ)2 and (u(q(x0 , t ), t ) − λ)2 < (ux (q (x0 , t ), t ) + λ)2 for all t ≥ 0. Suppose not, i.e. there exists a t0 such that (u(q(x0 , t ), t ) + λ)2 < (ux (q(x0 , t ), t ) + λ)2 and (u(q(x0 , t ), t ) − λ)2 < (ux (q(x0 , t ), t )+λ)2 on [0, t ), then (u(q(x0 , t ), t )+λ)2 = (ux (q(x0 , t ), t )+λ)2 or (u(q(x0 , t ), t )−λ)2 = (ux (q(x0 , t ), t )+λ)2 . Now, let 1 −q(x0 ,t ) q(x0 ,t ) ξ e y(ξ , t )dξ e 2 −∞

∫

I (t ) := and II (t ) :=

1 2

q(x0 ,t )

∫

∞

e

q(x0 ,t )

e−ξ y(ξ , t )dξ .

First, differentiating I (t ), we have dI (t ) dt

1

= − u(q(x0 , t ), t )e−q(x0 ,t )

q(x0 ,t )

∫

2

eξ y(ξ , t )dξ +

−∞

1 −q(x0 ,t ) q(x0 ,t ) ξ e e yt (ξ , t )dξ 2 −∞

∫

1 −q(x0 ,t ) q(x0 ,t ) ξ b−2 2 e e uyx + 2ux y + (u − u2x )x + λy dξ 2 2 2 −∞ 1 1 2 λ 2 ≥ u(ux − u)(q(x0 , t ), t ) + (u + ux − 2uux )(q(x0 , t ), t ) − (u − ux )(q(x0 , t ), t ) 2 4 2 1 2 λ 2 = (ux − u )(q(x0 , t ), t ) − (u − ux )(q(x0 , t ), t ) 4 2 1 1 2 = (ux + λ) (q(x0 , t ), t ) − (u + λ)2 (q(x0 , t ), t ) > 0, on [0, t0 ). 4 4

=

1



∫

u(ux − u)(q(x0 , t ), t ) −



(2.9)

Second, by the same argument, we get dII (t ) dt

= = ≤

1 2 1 2 1 2

u(q(x0 , t ), t )eq(x0 ,t )

∫

∞ q(x0 ,t )

u(ux + u)(q(x0 , t ), t ) −

1

u(ux + u)(q(x0 , t ), t ) −

1

2 4

e−ξ y(ξ , t )dξ + q(x0 ,t )

2

∞ −ξ

e

eq(x0 ,t )

∫

∞ q(x0 ,t )

e−ξ yt (ξ , t )dξ



e q(x0 ,t )

uyx + 2ux y +

(u2 + u2x + 2uux )(q(x0 , t ), t ) −

1

λ

4 1

2 1

4

4

= − (u2x − u2 )(q(x0 , t ), t ) −

∫

1

b−2

λ 2

2

(u − 2



) + λy dξ

u2x x

(ux + u)(q(x0 , t ), t )

(ux + u)(q(x0 , t ), t )

= − (ux (q(x0 , t ), t ) + λ)2 + (u(q(x0 , t ), t ) − λ)2 < 0,

on [0, t0 ).

(2.10)

Hence, it follows from (2.9), (2.10) and the continuity property of ODEs that

(ux (q(x0 , t ), t ) + λ)2 − (u(q(x0 , t ), t ) + λ)2 = −4I (t )(II (t ) + λ) > −4I (0)(II (0) + λ) > 0, and

(ux (q(x0 , t ), t ) + λ)2 − (u(q(x0 , t ), t ) − λ)2 = −4(I (t ) − λ)II (t ) > −4(I (0) − λ)II (0) > 0, for all t > 0, this implies that t0 can be extended to the infinity. Using (2.9) and (2.10) again, we have the following equation for [2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , t ), t ): d dt

[2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , t ), t ) = −4

d dt

[I (t )(II (t ) + λ)] − 4

d dt

[(I (t ) − λ)II (t )]

≥ −[(ux + λ)2 − (u + λ)2 ](q(x0 , t ), t )(II (t ) + λ) + [(ux + λ)2 − (u − λ)2 ](q(x0 , t ), t )I (t ) − [(ux + λ)2 − (u + λ)2 ](q(x0 , t ), t )II (t ) + [(ux + λ)2 − (u − λ)2 ](q(x0 , t ), t )(I (t ) − λ) = −λ[2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , t ), t ) − ux (q(x0 , t ), t )(2(ux (q(x0 , t ), t ) + λ)2 ) + 2(u + λ)2 II (t ) − 2(u − λ)2 I (t )

162

M. Zhu, Z. Jiang / Nonlinear Analysis: Real World Applications 13 (2012) 158–167

= −(ux (q(x0 , t ), t ) + λ)[2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , t ), t ) − ux (q(x0 , t ), t )[(u + λ)2 + (u − λ)2 ](q(x0 , t ), t ) + 2(u + λ)2 II (t ) − 2(u − λ)2 I (t ) ≥ −(ux (q(x0 , t ), t ) + λ)[2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , t ), t ),

(2.11)

where we use ux (q(x0 , t ), t ) = −I (t ) + II (t ). Now, recall (2.8), we get

∂t ux (q(x0 , t ), t )(q(x0 , t ), t ) ≤ =

1 2 1 4

u2 (q(x0 , t ), t ) −

1 2

u2x (q(x0 , t ), t ) − λux

[(u + λ)2 + (u − λ)2 − 2(ux + λ)2 ](q(x0 , t ), t ).

(2.12)

Substituting (2.12) into (2.11), it yields d dt

[2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , t ), t ) 1

[2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , t ), t ) ∫ t  × [2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , τ ), τ )dτ − 4u0x (x0 ) − 4λ . ≥

4

(2.13)

0

Before completing the proof, we need the following technical lemma. Lemma 2.2 ([2]). Suppose that Ψ (t ) is twice continuously differentiable satisfying



Ψ ′′ (t ) ≥ C0 Ψ ′ (t )Ψ (t ), Ψ (t ) > 0,

t > 0, C0 > 0, Ψ ′ (t ) > 0.

(2.14)

Then ψ(t ) blows up in finite time. Moreover, the blow up time can be estimated in terms of the initial datum as

 T ≤ max Let Ψ (t ) = with C0 =

1 . 4

t 0

 Ψ (0) . C0 Ψ (0) Ψ ′ (0) 2

,

[2(ux + λ)2 − (u + λ)2 − (u − λ)2 ](q(x0 , τ ), τ )dτ − 4u0x (x0 ) − 4λ, then (2.13) is an equation of type (2.14)

The proof is completed by applying Lemma 2.2.



Remark 2.1. Mckean got the necessary and sufficient condition for the Camassa–Holm equation in [7]. It is worth pointing out that Zhou and his collaborators [18] gave a new proof to Mckean’s theorem. However, the necessary and sufficient condition for (1.2) is still a challenging problem for us at present. Theorem 2.3. Let b = 2. Suppose that u0 ∈ H 2 (R) and there exists x0 ∈ R such that y0 (x0 ) = (1 − ∂x2 )u0 (x0 ) = 0, e−x0

∫

x0

eξ y0 (ξ )dξ > 2λ

ex0

and

∫

−∞

∞

e−ξ y0 (ξ )dξ < −2λ. x0

Then the corresponding solution u(x, t ) to Eq. (1.2) with u0 as the initial datum blows up in finite time. Proof. When b = 2, it is the famous Camassa–Holm equation with the weakly dissipative term. We can easily get 2

utx + uuxx − u +

1 2

u2x



2

+G∗ u +

1 2

u2x



+ λux = 0.

Differentiating ux at the point (q(x0 , t ), t ) with respect to t, we have d dt

ux (q(x0 , t ), t ) ≤

1 2

u2 (q(x0 , t ), t ) −

1 2

u2x (q(x0 , t ), t ) − λux (q(x0 , t ), t ) < 0.

The proof follows from Theorem 2.1 directly. So, to be concise, we omit the detailed proof. Remark 2.2. The condition in [13] is that u(0, t ) = 0 is odd,

∫

0

−∞

eξ y0 (ξ ) > 2λ and

∞

∫

e−ξ y0 (ξ ) < −2λ. 0

Theorem 2.3 is an improvement of that in [13].



M. Zhu, Z. Jiang / Nonlinear Analysis: Real World Applications 13 (2012) 158–167

163

3. Global existence Now, let us try to find a condition for global existence. Unfortunately, like the Degasperis–Procesi equation [11], only the following easy one can be proved at present. Theorem 3.1. Suppose that u0 ∈ H 3 (R), y0 = (1 − ∂x2 )u0 is have the same sign. Then the corresponding solution to (1.2) exists globally. Proof. Without loss of generality, we assume that y0 ≥ 0. It is sufficient to prove that ux (x, t ) has a lower and upper bound for all t. In fact, x 1 1 eξ y(ξ , t )dξ + ex ux (x, t ) = − e−x 2 2 −∞

∫

∞

∫

e−ξ y(ξ , t )dξ , x

so, x 1 x 1 ux (x, t ) ≥ − e−x eξ y(ξ , t )dξ ≥ − y(ξ , t )dξ 2 2 −∞ −∞ ∫ ∞ ∫ ∞ 1 1 y(ξ , t )dξ = − y0 (ξ , t )dξ ≥− 2 −∞ 2 −∞

∫

∫

and ux (x, t ) ≤

≤

1 2

x

∞

∫

e

1

e

−ξ

y(ξ , t )dξ ≤

x

∫

∞

2 −∞

y(ξ , t )dξ =

This completes the proof.

1

∫

1

∞

∫

2

y(ξ , t )dξ x

∞

2 −∞

y0 (ξ , t )dξ .



4. Persistence property Now, we shall investigate the following property for the strong solutions to (1.4) in L∞ -space with asymptotic exponential decay at infinity as their initial profiles. The main idea comes from a recent work of Zhou and his collaborators [6] for the standard Camassa–Holm equation (for slower decay rate, we refer to [19]). Theorem 4.1. Assume that for some T > 0 and s > satisfies that for some θ ∈ (0, 1),

5 , 2

u ∈ C ([0, T ]); H s (R) is a strong solution of (1.2) and u0 (x) = u(x, 0)

|u0 (x)|, |u0x (x)| ∼ O(e−θ x ). Then,

|u(x, t )|, |ux (x, t )| ∼ O(e−θ x ) uniformly in the time interval [0, T ]. Proof. First, we shall introduce the weight function to get the desired result. This function ϕN (x) with N ∈ Z+ is independent on t as follows:

ϕN (x) =

 1,

eθ x ,  θN e ,

x ≤ 0, x ∈ (0, N ), x ≥ N,

which implies that 0 ≤ ϕN′ (x) ≤ ϕN (x). From Eq. (1.4), we can get

∂t (uϕN ) + (uϕN )ux + ϕN ∂x G ∗



b 2

2

u +

3−b 2

u2x



+ λϕN u = 0.

Multiplying (4.1) by (uϕN )2p−1 with p ∈ Z+ and integrating the result in the x-variable, we get

∫

+∞

∫ +∞ ∂t (uϕN )(uϕN )2p−1 dx + (uϕN )ux (uϕN )2p−1 dx −∞ −∞   ∫ ∫ +∞ b 2 3−b 2 + ϕN ∂x G ∗ u + ux (uϕN )2p−1 + −∞

2

2

+∞

λϕN u(uϕN )2p−1 dx = 0, −∞

(4.1)

164

M. Zhu, Z. Jiang / Nonlinear Analysis: Real World Applications 13 (2012) 158–167

from which we can deduce that d dt

‖uϕN ‖L2p ≤ (‖ ‖

+ λ)‖uϕN ‖L2p

ux L∞

    3−b 2  b 2  + ϕN ∂x G ∗ u + ux   2

2

. L2p

Denoting M = supt ∈[0,T ] ‖u(t )‖H s and by Gronwall’s inequality, we obtain



‖uϕN ‖L2p ≤ ‖u0 ϕN ‖L2p

   ∫ t  b 2 3−b 2  (M +λ)t   + . ϕN ∂x G ∗ 2 u + 2 ux  2p dτ e 0 L

(4.2)

Taking the limits in (4.2), we get t

 ∫ ‖uϕN ‖L∞ ≤ ‖u0 ϕN ‖L∞ + 0

      ϕN ∂x G ∗ b u2 + 3 − b u2  dτ e(M +λ)t . x   2 2 L∞

(4.3)

Next differentiating (1.4) in the x-variable produces the equation utx + uuxx +

u2x

+∂

2 xG

 ∗

b 2

2

u +

3−b 2



u2x

+ λux = 0.

(4.4)

Using the weight function, we can rewrite (4.4) as

∂x (ux ϕN ) + uuxx ϕN + (ux ϕN )ux + ϕN ∂x2 G ∗



b 2

3−b

u2 +

2

u2x



+ λϕN ux = 0.

(4.5)

Multiplying (4.5) by (ux ϕN )2p−1 with p ∈ Z+ and integrating the result in the x-variable, it follows that +∞

∫

∂t (ux ϕN )(ux ϕN )

2p−1

∫

+∞

uuxx ϕN (ux ϕN )

2p−1

dx +

−∞ +∞

∫

ϕN ∂x2 G ∗



b 2

−∞

u2 +

3−b 2

(ux ϕN )ux (ux ϕN )2p−1 dx

dx +

−∞

+

+∞

∫

−∞

u2x



(ux ϕN )2p−1 dx +

∫

+∞

λϕN ux (ux ϕN )2p−1 dx = 0.

(4.6)

−∞

For the second term on the right side of (4.6), we know that

∫   

+∞

 

2p

uuxx ϕN (ux ϕN )2p−1 dx ≤ 2(‖u‖L∞ + ‖ux ‖L∞ )‖ux ϕN ‖L2p .

−∞

Using the above estimate and Hölder inequality, we deduce that d dt

‖ux ϕN ‖L2p ≤ (2M + λ)‖ux ϕN ‖L2p + ‖ϕN ∂x S ‖L2p .

Thanks to Gronwall’s inequality, it holds that



‖ux ϕN ‖L2p ≤ ‖u0x ϕN ‖L2p

   ∫ t  b 2 3−b 2  2 (2M +λ)t   + . ϕN ∂x G ∗ 2 u + 2 ux  2p dτ e 0 L

(4.7)

Taking the limits in (4.7), we have

 ∫ ‖ux ϕN ‖L∞ ≤ ‖u0x ϕN ‖L∞ + 0

t

      ϕN ∂ 2 G ∗ b u2 + 3 − b u2  dτ e(2M +λ)t . x x   2 2 L∞

(4.8)

Combining (4.3) and (4.8) together, it follows that

‖uϕN ‖

L∞

+ ‖ux ϕN ‖

L∞

≤ (‖u0 ϕN ‖

L∞

+ ‖u0x ϕN ‖

L∞

)e

(2M +λ)t

+e

(2M +λ)t

∫ t      ϕN ∂x G ∗ b u2 + 3 − b u2  x   2 2 0 L∞

     b 2 3−b 2  2   + ϕN ∂x G ∗ u + ux  dτ . 2 2 L∞ A simple calculation shows that there exists c0 > 0, depending only on θ ∈ (0, 1), such that for any N ∈ Z+ ,

ϕN (x)

∫

∞

e−|x−y| −∞

1

ϕN (y)

dy ≤ c0 =

4 1−θ

.

(4.9)

M. Zhu, Z. Jiang / Nonlinear Analysis: Real World Applications 13 (2012) 158–167

165

Thus, for any appropriate function f and g, one sees that

|ϕN G ∗ f (x)g (x)| = ≤ ≤ ≤

  ∫ ∞  1 −|x−y|  ϕN (x) e f (y)g (y)dy 2 −∞ ∫ ∞ 1 1 e−|x−y| ϕN (x) ϕN (y)f (y)g (y)dy 2 ϕ N (y) −∞   ∫ ∞ 1 1 e−|x−y| ϕN (x) dy ‖ϕN f ‖L∞ ‖g ‖L∞ 2 ϕN (y) −∞ c0 ‖ϕN f ‖L∞ ‖g ‖L∞ .

Similarly, we can get

|ϕN ∂x G ∗ f (x)g (x)| ≤ c0 ‖ϕN f ‖L∞ ‖g ‖L∞ , and

|ϕN ∂x2 G ∗ f (x)g (x)| ≤ c0 ‖ϕN f ‖L∞ ‖g ‖L∞ . Thus, inserting the above estimates into (4.9), there exists a constant c˜ = c˜ (M , T , λ) ≥ 0 such that

‖uϕN ‖L∞ + ‖ux ϕN ‖L∞ ≤ c˜ (‖u0 ϕN ‖L∞ + ‖u0x ϕN ‖L∞ ) + c˜

t

∫

(‖u‖L∞ + ‖ux ‖L∞ )(‖uϕN ‖L∞ + ‖ux ϕN ‖L∞ )dτ 0

≤ c˜ (‖u0 ϕN ‖L∞ + ‖u0x ϕN ‖L∞ ) + c˜

t

∫

(‖uϕN ‖L∞ + ‖ux ϕN ‖L∞ )dτ . 0

Hence, for any t ∈ Z+ and any t ∈ [0, T ], we have

‖uϕN ‖L∞ + ‖ux ϕN ‖L∞ ≤ c˜ (‖u0 ϕN ‖L∞ + ‖u0x ϕN ‖L∞ ) ≤ c˜ (‖u0 max(1, eθ x )‖L∞ + ‖u0x max(1, eθ x )‖L∞ ).

(4.10)

Finally, taking the limit as N goes to infinity in (4.10) we find that for any t ∈ [0, T ]

‖ueθ x ‖L∞ + ‖ux eθ x ‖L∞ ≤ c˜ (‖u0 max(1, eθ x )‖L∞ + ‖u0x max(1, eθ x )‖L∞ ), which completes the proof of the theorem.



5. Infinite propagation speed Recently, Zhou and his collaborators established the infinite propagation speed for the Camassa–Holm equation in [6]. Later, Guo [13,14] considered a similar problem on the weakly dissipative Camassa–Holm equation and the weakly dissipative Degasperis–Procesi equation. The purpose of this section is to give a more detailed description on the corresponding strong solution u(x, t ) to (1.2) in its life span with initial data u0 (x) being compactly supported. The main theorem reads as follows. Theorem 5.1. Let 0 ≤ b ≤ 3. Assume that for some T ≥ 0 and s ≥ 52 , u ∈ C ([0, T ]; H s (R)) is a strong solution of (1.2). If u0 (x) = u(x, 0) has compact support [a, c ], then for t ∈ (0, T ], we have u(x, t ) =

f+ (t )e−x , f− (t )e−x ,



for x > q(c , t ) for x < q(a, t ),

where f+ (t ) and f− (t ) denote continuous nonvanishing functions, with f+ (t ) > 0 and f− (t ) < 0 for t ∈ (0, T ]. Furthermore, f+ (t ) is a strictly increasing function, while f− (t ) is a strictly decreasing function. Proof. Since u0 has compact support in x in the interval [a, c ] from (2.6), so does y(, t ) in x in the interval [q(a, t ), q(c , t )] in its lifespan. Hence the following functions are well-defined. E (t ) =

∫

ex y(x, t )dx and

F (t ) =

∫

R

e−x y(x, t )dx, R

with

∫

ex y0 (x)dx = 0

E0 = R

∫ and

e−x y0 (x)dx = 0.

F0 = R

166

M. Zhu, Z. Jiang / Nonlinear Analysis: Real World Applications 13 (2012) 158–167

Then, for x > q(c , t ), we have 1 −|x| 1 e ∗ y(x, t ) = e−x 2 2

u(x, t ) =

q(b,t )

∫

q(a,t )

eτ y(τ , t )dτ =

1 −x e E (t ).

(5.1)

2

Similarly, when x < q(a, t ), we get 1 1 u(x, t ) = e−|x| ∗ y(x, t ) = ex 2 2

q(b,t )

∫

q(a,t )

e−τ y(τ , t )dτ =

1 2

ex F (t ).

(5.2)

Hence, as consequences of (5.1) and (5.2), we have 1 −x e E (t ), 2

u(x, t ) = −ux (x, t ) = uxx (x, t ) =

as x > q(c , t )

(5.3)

and u(x, t ) = ux (x, t ) = uxx (x, t ) =

1 2

ex F (t ),

as x < q(a, t ).

(5.4)

On the other hand, dE (t ) dt

∫

ex yt (x, t )dx.

= R

It is easy to get yt = −uux + (uux )xx − ∂x



b 2

Substituting the identity (5.5) into dE (t ) dt

u2 +

dE (t ) , dt

3−b 2

u2x



− λu + λuxx .

(5.5)

we obtain

  ∫ b 2 3−b 2 −uux + (uux )xx − ∂x u + ux dx + ex (−λu + λuxx )dx 2 2 R R   ∫ 3−b 2 b 2 x u + ux dx, = e ∫

ex

=



2

R

2

where we use (5.3) and (5.4). Therefore, in the lifespan of the solution, we have E (t ) =

∫ t∫ 0

ex R



b 2

u2 +

3−b 2

u2x



(x, τ )dxdτ > 0.

By the same argument, one can check that the following identity for F (t ) is true F (t ) = −

∫ t∫ 0

e−x R



b 2

u2 +

3−b 2

u2x



(x, τ )dxdτ < 0.

In order to complete the proof, it is sufficient to let f+ (t ) =

1 E 2

(t ) and f− (t ) = 12 F (t ).



Remark 5.1. Theorem 4.1 means that no matter what the profile of the compactly supported initial datum u0 (x) is (no matter whether it is positive or negative), for any t > 0 in its lifespan, the solution u(x, t ) is positive at infinity and negative at negative infinity. Acknowledgments This work is partially supported by the Zhejiang Innovation Project (T200905) and the ZJNSF (Grant No. R6090109). References [1] D. Holm, M. Staley, Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a 1 + 1 nonlinear evolutionary PDE, Phys. Lett. A 308 (2003) 437–444. [2] Y. Zhou, On solutions to the Holm–Staley b-family of equations, Nonlinearity 23 (2010) 369–381. [3] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661–1664. [4] A. Degasperis, M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Scientific, Singapore, 1999, pp. 23–37. [5] A. Constantin, J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasilinear hyperbolic equation, Commun. Pure Appl. Math. 51 (1998) 475–504.

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[6] A. Himonas, G. Misiolek, G. Ponce, Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa–Holm equation, Comm. Math. Phys. 271 (2007) 511–522. [7] H.P. Mckean, Breakdown of a shallow water equation, Asian J. Math. 2 (1998) 767–774. [8] L. Molinet, On well-posedness results for Camassa–Holm equation on the line: a survey, J. Nonlinear Math. Phys. 11 (4) (2004) 521–533. [9] Y. Zhou, Wave breaking for a shallow water equation, Nonlinear Anal. 57 (2004) 137–152. [10] Y. Zhou, Wave breaking for a periodic shallow water equation, J. Math. Anal. Appl. 290 (2004) 591–604. [11] Y. Zhou, Blow up phenomena for the integrable Degasperis–Procesi equation, Phys. Lett. A 328 (2004) 157–162. [12] W. Niu, S. Zhang, Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation, J. Math. Anal. Appl. 374 (2011) 166–177. [13] Z. Guo, Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa–Holm equation, J. Math. Phys. 49 (2008) 033516. [14] Z. Guo, Some properties of solutions to the weakly dissipative Degasperis–Procesi equation, J. Differential Equations 246 (2009) 4332–4344. [15] L. Ni, Y. Zhou, Wave breaking and propagation speed for a class of nonlocal dispersive θ -equations, Nonlinear Anal. RWA 12 (2011) 592–600. [16] Zhou Yong, Blow-up of solutions to the DGH equation, J. Funct. Anal. 250 (1) (2007) 227–248. [17] Zhou Yong, Blow-up of solutions to a nonlinear dispersive rod equation, Calc. Var. Partial Differ. Equ. 25 (1) (2006) 63–77. [18] Z. Jiang, L. Ni, Y. Zhou, Wave breaking for the Camassa–Holm equation, Preprint, 2010. [19] L. Ni, Y. Zhou, A new asymptotic behavior of solutions to the Camassa–Holm equation, Proc. Amer. Math. Soc (2011) (in press). Article electronically published on May 12, 2011.