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^{2}

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We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

In this paper, we consider the Cauchy problem of the following weakly dissipative modified two-component Camassa-Holm system:

The Camassa-Holm equation [

Since the system was derived physically by Constantin and Ivanov [

Recently, the CH2 system was generalized into the following modified two-component Camassa-Holm (MCH2) system:

In general, it is quite difficult to avoid energy dissipation mechanisms in a real world. Ghidaglia [

This paper is organized as follows: In Section

In this section, by applying Kato’s semigroup theory [

First, we introduce some notations. All spaces of functions are assumed to be over

For convenience, we state here Kato’s theorem in the form suitable for our purpose.

Consider the following abstract quasilinear evolution equation:

Let

Assume that

and

where,

Moreover, the map

We now provide the framework in which we will reformulate system (

Note that if

Given

Moreover, the solution depends continuously on the initial data, that is, the mapping

The remainder of this section is devoted to the proof of Theorem

Let

Set

We break the argument into several lemmas.

The operator

The operator

The operator

The proof of the above five lemmas can be done similarly as in [

Hence, according to Kato’s theorem (Theorem

Let

Then

Let

This proves (a). Taking

Next, we prove (b). Note that

This proves (b) and completes the proof of the Lemma

Combining Theorem

In this section, we present the precise blow-up scenario and the blow-up rate for strong solutions to (

Let

Denote

In view of the identity

Therefore, an integration by parts yields

Thus, the statement of the conservation law follows.

For every

So, if

Let

The proof of the theorem is similar to the proof of Theorem 3.1 in [

Consider the following differential equation equation:

Let

The following result is proved only with regard to

We now present a precise blow-up scenario for strong solutions to (

Let

Multiplying the first equation in (

Repeating the same procedure to the second equation in (

A combination of (

Differentiating the first equation in (

Similarly,

A combination of (

Assume that there exists

Therefore,

The above discussion shows that if there exist

This inequality, Sobolev’s embedding theorem and Theorem

On the other hand, we see that if

Let

The function

Let

Applying Theorems

Define now

Evaluating (

This relation together with (

Choose

Since

Note that

It follows from the above inequality that

The obtained contradiction completes the proof of the relation (

For

Since

By the arbitrariness of

In this section, we discuss the blow-up phenomena of (

Let

As mentioned earlier, here we only need to show that the above theorem holds for

Define now

Clearly

Inequality (

Take

It then follows that

Note that if

Due to

Let

In view of (

Note that

Thus,

Using the following inequality:

Taking

Note that if

Since

Applying Theorem

This work was partially supported by NSF of China (11071266), Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, and the Educational Science Foundation of Chongqing China (KJ121302).