Well-Posedness and Infinite Propagation Speed for the Fifth-Order Camassa-Holm Equation with Weakly Dissipative Term ()
1. Introduction
In 1993, Camassa and Holm [1] derived an integrable shallow water equation taking the form as
(1.1)
where
denotes the height of the water above the flat bottom. Equation (1.1) describes wave motion in shallow water regime. Due to its mathematical importance and wide applications in physics, such equation has been intensively studied in the past decades and many interesting results have been obtained. Local well-posedness for the initial datum in
with
was proved in [2] [3] . The blow-up phenomenon was studied in [2] - [7] . McKean [7] (see also [6] for a simple proof) proved that the Camassa-Holm equation breaks if and only if some portion of the positive part of
lies to the left of some portion of its negative part, here
. The hierarchy properties and algebro-geometric solutions of the Camassa-Holm equation were proposed in [8] . Global weak solution was studied in [9] [10] [11] .
Later, researchers are concerned with the Camassa-Holm equation with weakly dissipative term
(1.2)
Wu and Yin proved in [12] [13] that the solution of (1.2) decays to zero as time goes to infinity if the initial momentum density
does not change its sign. Unlike the Camassa-Holm equation, there are no traveling wave solutions of (1.2) (see [12] ). Global existence and blow-up phenomena were studied in [12] - [17] .
In recent years, the following fifth-order Camassa-Holm equation was studied by Liu and Qiao [18] :
(1.3)
where
are real constants. In [18] , the authors studied peakon solutions including single pseudo-peakons, two-peakons and three-peakons interactional solutions of (1.3). In [19] , Tang and Liu proved that the Cauchy problem of this equation is locally well-posed in the critical Besov space
or in
with
,
when
. Zhu, Cao, Jiang and Qiao [20] studied the global existence of the solution of (1.3) with
. For more mathematical studies of (1.3), we refer to [21] [22] [23] [24] [25] .
So far, there have been many researches on Camassa-Holm equation. However, the results on the fifth-order Camassa-Holm equation are few. In this paper, we consider the Cauchy problem of the fifth-order Camassa-Holm equation with weakly dissipative term
(1.4)
(1.5)
(1.6)
where
is a constant and
are two parameters satisfying
. Our main purpose is to study the existence of global solution for this problem and investigate the properties of the solution.
Throughout the paper, we denote by
the Lebesgue space equipped with the norm
and
For
,
denotes the nonhomogeneous Sobolev space defined by
where
is the Fourier transform of u.
This paper is organized as follows. In Section 2, we present the local well-posedness result and the blow up criterion. In Section 3, we discuss the problem of global existence. Finally, we consider the infinite propagation speed in Section 4.
2. Blow up Criterion
In this section, we discuss the blow up criterion for the solution to (1.4) - (1.6). To this aim, we first give the following existence theorem.
Theorem 2.1. Let
with
. Then the Cauchy problem (1.4) - (1.6) admits a unique solution
such that
where
.
Theorem 2.1 can be proved by applying Kato’s method [26] . Since the argument is standard, we omit the details for simplicity. If
and
we say the solution blows up in finite time. If the norm
is bounded at any large time, we say the solution exists globally.
Define
(2.1)
then the function
can be expressed as
Now we state the main result of this section.
Theorem 2.2. Assume that
, and T is the maximal existence time of the solution obtained by Theorem 2.1.
1) If
, then the solution blows up in finite time if and only if
(2.2)
2) If
, then the solution blows up in finite time if and only if
(2.3)
Proof: Using (1.5), we obtain
then
(2.4)
In case of
, we have
which implies
If
, then
(2.5)
By Gronwall’s inequality, there holds
(2.6)
Hence,
is bounded for all
, which contradicts the maximal property of T.
In case of
, we can obtain similar result with the same arguments as above. This completes the proof of Theorem 2.2.
3. Global Existence Results
In this section, we will show two global existence results. To begin with, motivated by Mckean’s deep observation for the Camassa-Holm equation [7] , we define the particle trajectory by
(3.1)
where T is the lifespan of the solution. From (3.1), we get
which yields
(3.2)
The relation (3.2) implies that
is always positive, hence
is an increasing function. A direct calculation gives
so we have
(3.3)
From (3.3), we also get
(3.4)
Theorem 3.1. Assume
or
, and
with
, then the solution of (1.4) - (1.6) exists globally in time.
Proof: We first consider the case
. Multiplying both sides of Equation (1.4) by y and integrating with respect to x, we have
hence, we obtain
. By (2.4) and Sobolev’s inequality, we see that
is bounded by
, so we can extend the local solution to be a global solution.
Then, we give the proof for
. Let
then
The above identity gives
, so once again we can see
is bounded. This ends the proof of Theorem 3.1.
Theorem 3.2. Let
and
with
. Assume that
does not change sign, then the solution of (1.4) - (1.6) exists globally in time.
Proof: Differentiating
with respect to t and using (1.5), we have
which yields
(3.5)
When
, using the expression of G (see (2.1)),
and
can be represented as
(3.6)
and
(3.7)
Without loss of generality, we may assume
as the case
can be discussed similarly. If
, by (3.5) and (3.7), we have
and
Similarly, if
, we can also obtain
Hence, by Theorems 2.1 - 2.2, we know that the problem (1.4) - (1.6) has a unique global solution.
4. Infinite Propagation Speed
In this section, we study the infinite propagation phenomenon for the Equation (1.4). To this aim, we set
and
The main result of this section is stated as follows.
Theorem 4.1. Let
. Suppose that the initial value
is supported in the interval
. Then for any
, the solution
of (1.4) - (1.6) can be expressed by
Moreover, if
, then
is a strictly increasing function and
is a strictly decreasing function. Similarly, if
, then
is a strictly increasing function and
is a strictly decreasing function.
Proof: By (3.3), we have
Since
(4.1)
when
, we get
Similarly, when
, we have
Note that
(4.2)
By
then we get
(4.3)
Inserting (4.3) into (4.2), we have
then
(4.4)
Applying similar arguments as above, we can obtain
(4.5)
According to (4.4) and (4.5), it is easy to see that if
,
and
, the function
is strictly increasing and the function
is strictly decreasing. In the same way, we can obtain the desired result when
,
and
.
Acknowledgements
The author is extremely appreciative of the reviewers valuable suggestions and comments on the manuscript.