Blow-up phenomena and global existence for the weakly dissipative generalized periodic Degasperis-Procesi equation

In this paper, we investigate the Cauchy problem of a weakly dissipative generalized periodic Degasperis-Procesi equation. The precise blow-up scenarios of strong solutions to the equation are derived by a direct method. Several new criteria guaranteeing the blow-up of strong solutions are presented. The exact blow-up rates of strong solutions are also determined. Finally, we give a new global existence results to the equation.

MSC:35G25, 35Q35, 58D05.

Recently, the following generalized periodic Degasperis-Procesi equation (μ DP) was introduced and studied in [1–3]

where u(t,x) is a time-dependent function on the unite circle \mathbb{S}=\mathbb{R}/\mathbb{Z} and \mu (u)={\int }_{\mathbb{S}}u(t,x)dx denotes its mean. The μ DP equation can be formally described as an evolution equation on the space of tensor densities over the Lie algebra of smooth vector fields on the circle \mathbb{S}. In [2], the authors verified that the periodic μ DP equation describes the geodesic flows of a right-invariant affine connection on the Fréchet Lie group {Diff}^{\mathrm{\infty }}(\mathbb{S}) of all smooth and orientation-preserving diffeomorphisms of the circle \mathbb{S}.

Analogous to the generalized periodic Camassa-Holm (μ CH) equation [4–6], μ DP equation possesses bi-Hamiltonian form and infinitely many conservation laws. Here we list some of the simplest conserved quantities:

where y=\mu (u)-u_{xx}, A=\mu -{\partial }_x^2 is an isomorphism between H^S and H^{s-1}. Moreover, it is easy to see that {\int }_{\mathbb{S}}u(t,x)dx is also a conserved quantity for the μ DP equation.

Obviously, under the constraint of \mu \equiv 0, the μ DP equation is reduced to the μ Burgers equation [7].

It is clear that the closest relatives of the μ DP equation are the DP equation [8–11]

which was derived by Degasperis and Procesi in [8] as a model for the motion of shallow water waves, and its asymptotic accuracy is the same as for the Camassa-Holm equation.

Generally speaking, energy dissipation is a very common phenomenon in the real world. It is interesting for us to study this kind of equation. Recently, Wu and Yin [12] considered the weakly dissipative Degasperis-Procesi equation. For related studies, we refer to [13] and [14]. Liu and Yin [15] discussed the blow-up, global existence for the weakly dissipative μ-Hunter-Saxton equation.

In this paper, we investigate the Cauchy problem of the following weakly dissipative periodic Degasperis-Procesi equation [16]:

the constant λ is a nonnegative dissipative parameter and the term \lambda y=\lambda (\mu (u)-u_{xx}) models energy dissipation. Obviously, if \lambda =0 then the equation reduces to the μ DP equation. we can rewrite the system (1.1) as follows:

Let G(x):=\frac{1}{2}{x}^2-\frac{1}{2}|x|+\frac{13}{12}, x\in \mathbb{R} be the associated Green’s function of the operator A^{-1}, then the operator can be expressed by its associated Green’s function,

where ∗ denotes the spatial convolution. Then equation (1.1) takes the equivalent form of a quasi-linear evolution equation of hyperbolic type:

It is easy to check that the operator A=\mu -{\partial }_x^2 has the inverse

Since A^{-1} and {\partial }_x commute, the following identities hold:

and

The paper is organized as follows. In Section 2, we briefly give some needed results, including the local well-posedness of equation (1.1), and some useful lemmas and results which will be used in subsequent sections. In Section 3, we establish the precise blow-up scenarios and blow-up criteria of strong solutions. In Section 4, we give the blow-up rate of strong solutions. In Section 5, we give two global existence results of strong solutions.

Remark 1.1 Although blow-up criteria and global existence results of strong solutions to equation (1.1) are presented in [16], our blow-up results improve considerably earlier results.

In this section we recall some elementary results which we want to use in this paper. We list them and skip their proofs for conciseness. Local well-posedness for equation (1.1) can be obtained by Kato’s theory [17], in [16] the authors gave a detailed description on well-posedness theorem.

Theorem 2.1 [16]

Let s>3/2 and u_0\in H^s(\mathbb{S}); then there is a maximal time T and a unique solution

of the Cauchy problems (1.1) which depends continuously on the initial data, i.e. the mapping

is continuous.

Remark 2.1 The maximal time of existence T>0 in Theorem 2.1 is independent of the Sobolev index s>3/2.

Next we present the Sobolev-type inequalities, which play a key role to obtain blow-up results for the Cauchy problem (1.1) in the sequel.

Lemma 2.2 [18]

If f\in H^1(\mathbb{S}) is such that {\int }_{\mathbb{S}}f(x)dx=0, then we have

Lemma 2.3 [19]

If r>0, let \mathrm{\Lambda }={(1-{\partial }_x^2)}^{1/2}, then

where c is a constant depending only on r.

Lemma 2.4 [20]

Let t_0>0 and v\in C^1([0,t_0);H^2(\mathbb{R})), then for every t\in [0,t_0) there exists at least one point \xi (t)\in \mathbb{R} with

and the function m is almost everywhere differentiable on (0,t_0) with

We also need to introduce the classical particle trajectory method which is motivated by McKean’s deep observation for the Camassa-Holm equation in [21]. Suppose u(x,t) is the solution of the Camassa-Holm equation and q(x,t) satisfies the following equation:

where T is the maximal existence time of solution, then q(t,\cdot ) is a diffeomorphism of the line. Taking the derivative with respect to x, we have

Hence

which is always positive before the blow-up time.

In addition, integrating both sides of the first equation in equation (1.1) with respect to x on \mathbb{S}, we obtain

it follows that

where

In this section, we are able to derive an import estimate for the L^{\mathrm{\infty }}-norm of strong solutions. This enables us to establish precise blow-up scenario and several blow-up results for equation (1.1).

Lemma 3.1 Let u_0\in H^s, s>3/2 be given and assume the T is the maximal existence time of the corresponding solution u to equation (1.1) with the initial data u_0. Then we have

Proof The first equation of the Cauchy problem (1.1) is

In view of equation (1.5), we have

A direct computation implies that

It follows that

So we have

In view of equation (2.1) we have

Combing the above relations, we arrive at

Integrating the above inequality with respect to on [0,t] yields

Thus

In view of the diffeomorphism property of q(t,\cdot ), we can obtain

This completes the proof of Lemma 3.1. □

Theorem 3.2 Let u_0\in H^s, s>3/2 be given and assume that T is the maximal existence time of the corresponding solution u(t,x) to the Cauchy problem (1.1) with the initial data u_0. If there exists M>0 such that

then the H^s-norm of u(t,\cdot ) does not blow up on [0,T).

Proof We assume that c is a generic positive constant depending only on s. Let \mathrm{\Lambda }={(1-{\partial }_x^2)}^{1/2}. Applying the operator {\mathrm{\Lambda }}^s to the first one in equation (1.3), multiplying by {\mathrm{\Lambda }}^{s}u, and integrating over \mathbb{S}, we obtain

Let us estimate the first term of the above equation,

where we used Lemma 2.3 with r=s. Furthermore, we estimate the second term of the right hand side of equation (3.3) in the following way:

Combing equations (3.4) and (3.5) with equation (3.3) we arrive at

An application of Gronwall’s inequality and the assumption of the theorem yield

This completes the proof of the theorem. □

The following result describes the precise blow-up scenario. Although the result which is proved in [16], our method is new, concise, and direct.

Theorem 3.3 Let u_0\in H^s, s>3/2 be given and assume that T is the maximal existence time of the corresponding solution u(t,x) to the Cauchy problem (1.1) with the initial data u_0. Then the corresponding solution blows up in finite time if and only if

Proof Since the maximal existence time T is independent of the choice of s by Theorem 2.1, applying a simple density argument, we only need to consider the case s=3. Multiplying the first one in equation (1.2) by y and integrating over \mathbb{S} with respect to x yield

If u_x is bounded from below on [0,T)\times \mathbb{S}, then there exists N>\lambda >0 such that

then

Applying Gronwall’s inequality then yields for t\in [0,T)

Note that

Since u_x\in H^2\subset H^1 and {\int }_{\mathbb{S}}{u}_x=0, Lemma 2.2 implies that

Theorem 3.1 ensures that the solution u does not blow up in finite time. On the other hand, by the Sobolev embedding theorem it is clear that if

then . This completes the proof of the theorem. □

We now give first sufficient conditions to guarantee wave breaking.

Theorem 3.4 Let u_0\in H^s, s>3/2 and T be the maximal time of the solution u(t,x) to equation (1.1) with the initial data u_0. If

then the corresponding solution to equation (1.1) blow up in finite time in the following sense: there exists T_0 satisfying

where \alpha =3|{\mu }_0|(\frac{3|{\mu }_0|(\frac{1}{2}|{\mu }_0|+2{\mu }_2)}{\lambda }+{\parallel u_0\parallel }_{L^{\mathrm{\infty }}}), such that

Proof As mentioned early, we only need to consider the case s=3. Let

and let \xi (t)\in \mathbb{S} be a point where this minimum is attained by using Lemma 2.4. It follows that

Differentiating the first one in equation (1.3) with respect to x, we have

From equation (1.6) we deduce that

Obviously u_{xx}(t,\xi (t))=0 and u(t,\cdot )\in H^3(\mathbb{S})\subset C^2(\mathbb{S}). Substituting (t,\xi (t)) into equation (3.6), we get

Set

Then we obtain

Note that if , then for all t\in [0,T). From the above inequality we obtain

Since

then there exists T_0,

such that {lim}_{t\to T_0}m(t)=-\mathrm{\infty }. Theorem 3.3 implies that the solution u blows up in finite time. □

We give another blow-up result for the solutions of equation (1.1).

Theorem 3.5 Let u_0\in H^s, s>3/2 and T be the maximal time of the solution u(t,x) to equation (1.1) with the initial data u_0. If u_0 is odd satisfies , then the corresponding solution to equation (1.1) blows up in finite time.

Proof By \mu (u(t,-x))={\mu }_0(t,-x)e^{-\lambda t}=-{\mu }_0(t,x)e^{-\lambda t}=-\mu (u(t,x)), we can check the function

is also a solution of equation (1.1), therefore u(x,t) is odd for any t\in [0,T). By continuity with respect to x of u and u_{xx}, we get

Define h(t):=u_x(t,0) for t\in [0,T). From equation (3.6), we obtain

Note that if , then for all t\in [0,T). From the above inequality we obtain

Since

there exists T_0,

such that {lim}_{t\to T_0}m(t)=-\mathrm{\infty }. Theorem 3.3 implies that the solution u blows up in finite time. □

In this section, we consider the blow-up profile; the blow-up rate of equation (1.1) with respect to time can be shown as follows.

Theorem 4.1 Let u_0\in H^s, s>3/2 and T be the maximal time of the solution u(t,x) to equation (1.1) with the initial data u_0. If T is finite, then

Proof It is inferred from Lemma 2.4 that the function

is locally Lipschitz with , t\in [0,T). Note that u_{xx}=0, a.e. t\in [0,T). Then we deduce that

It follows that

Thus,

Now fix any \epsilon \in (0,1). In view of Theorem 3.1, there exists t_0\in (0,T) such that . Being locally Lipschitz, the function m(t) is absolutely continuous on [0,T). It then follows from the above inequality that m(t) is decreasing on [t_0,T) and satisfies

Since m(t) is decreasing on [t_0,T), it follows that

It is found from equation (4.1) that

Integrating both sides of equation (4.2) on (t,T), we obtain

that is,

By the arbitrariness of \epsilon \in (0,\frac{1}{2}), we have

This completes the proof of the theorem. □

In this section, we will present some global existence results. Let us now prove the following lemma.

Lemma 5.1 Let u_0\in H^s, s>3/2 be given and assume that T>0 is the maximal existence time of the corresponding solution u(t,x) to the Cauchy problem (1.1). Let q\in C^1([0,T)\times \mathbb{R};\mathbb{R}) be the unique solution of equation (2.1). Then we have

where y=\mu (u)-u_{xx}.

Proof By the first one in equation (1.2) and equation (2.1) we have

Therefore

 □

Lemma 5.1 and equation (2.2) imply that y and y_0 have the same sign.

Theorem 5.2 Let u_0\in H^s, s>3/2. If y_0={\mu }_0-u_{0,xx}\in H^1 does not change sign, then the corresponding solution u(t,x) to equation (1.1) with the initial data u_0 exists globally in time.

Proof By equation (2.1), we know that q(t,\cdot ) is diffeomorphism of the line and the periodicity of u with respect to spatial variable x, given t\in [0,T), there exists a \xi (t)\in \mathbb{S} such that u_x(t,\xi (t))=0.

We first consider the case that y_0\ge 0 on \mathbb{S}, in which case Lemma 5.1 ensures that y\ge 0. For x\in [\xi (t),\xi (t)+1], we have

It follows that u_x(t,x)\ge -|{\mu }_0|.

On the other hand, if y_0\le 0 on \mathbb{S}, then Lemma 5.1 ensures that y\le 0. Therefore, for x\in [\xi (t),\xi (t)+1], we have

It follows that u_x(t,x)\ge -|{\mu }_0|. By using Theorem 3.2, we immediately conclude that the solution is global. This completes the proof of the theorem. □

Corollary 5.3 If the initial value u_0\in H^3 such that

then the corresponding solution u of the initial value u_0 exists globally in time.

Proof Since {\int }_{\mathbb{S}}{\partial }_x^2{u}_{0}dx=0, by Lemma 2.2, we obtain

If {\mu }_0\ge 0, we have

If , we have

 □

Thus the theorem is proved by using Theorem 5.2.

This work is partially supported by the NSFC (Grant No. 11101376) the HiCi Project (Grant No. 27-130-35-HiCi).

Affiliations

1. Department of Mathematics, Tianshui Normal University, Tianshui, Gansu, 741001, P.R. China

   • Caochuan Ma

2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

   • Ahmed Alsaedi
   • , Tasawar Hayat
   •  & Yong Zhou

3. Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad, 44000, Pakistan

   • Tasawar Hayat

4. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, P.R. China

   • Yong Zhou

Corresponding author

Correspondence to Caochuan Ma.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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