Abstract

We investigate the Cauchy problem for the modified Novikov equation. We establish blow-up criteria on the initial data to guarantee the corresponding solution blowing up in finite time.

1. Introduction

In this paper, we consider the following Cauchy problem of the modified Novikov equation: where the coefficients and are positive constants.

In [1], Lai et al. presented the global existence of strong solutions and gave a blow-up scenario of strong solutions to the equation.

By using Green’s function for the operator , (1) is equivalent to the nonlocal equation where notation denotes the spatial convolution.

Letting and using the scaling translation , (1) can be reformulated into the Novikov equation which was derived by Novikov in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity [2]; subsequently, he found a scalar Lax pair for the Novikov equation (also see [3]) and proved that the Novikov equation is integrable. The equation has been investigated by many scholars. Hone and Wang gave a matrix Lax pair for the Novikov equation in [4] and showed how it was related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. By using the matrix Lax pair, Hone et al. calculated the explicit formulas for multipeak on solutions of (1) in [3]. Ni and Zhou showed that the Novikov equation is well-posed in by applying Kato’s semigroup theory and the Novikov equation is locally well-posed in the Besov spaces with the critical index and also considered the persistence properties of the solution. In [5], Jiang and Ni gave sufficient conditions on the initial data to guarantee the formulation of singularities in finite time and a global existence result was also established in [6]. It is worth pointing out recent many works have been done for the Novikov equation and the related equations, one can refer to [7–12] and the references therein.

Now, we give some elementary results and a blow-up scenario of strong solutions which will be used in this paper.

Theorem 1 (see [13]). Given with , then there exist a maximal and a unique solution to (1) such that Moreover, the solution depends continuously on the initial data; that is, the mapping is continuous.

Theorem 2 (see [1]). Assume , , and let be the maximal existence time of the solution to (1) with the initial data . If , then the corresponding solution blows up in finite time if and only if

We also need to introduce the classical particle trajectory method. Suppose is a solution of the Novikov equation; let be the particle line evolved by the solution : Then which is always positive before the blow-up time. Therefore, the function is an increasing diffeomorphism of the line before blow-up.

Let ; the following identity can be obtained: In fact, direct computation yields

Remark 3. From (8), it follows that if then . Since , for , therefore, we obtain . If , the result is similar.

2. Blow-Up Criteria

In this section, we present the following blow-up criteria on the initial data to guarantee that the corresponding strong solution of (1) blowing up. Our method is partially motivated by [14].

Theorem 4. If , suppose that , and there exists such that and , Then the corresponding solution to the modified Novikov equation (1) with as the initial datum blows up in finite time.

Proof. By the local well-posedness theorem and a density argument, it suffices to consider the case ; without loss of generality, we take for simplicity of notation. We also assume ; otherwise, solutions are trivial.
Suppose that the solution exists globally. Due to (8) and the initial condition (10), we have for all . Since , , , we can write and as follows: As a result of (12) for all .
From (12) and differentiating with respect to , we have Equation (1) can be rewritten as
Firstly, we can estimate the first term as
We also apply the following inequality in [6]: So we can derive
Putting (19) into (17), we have
Similarly, we have
Putting (21) and (22) into (15), we obtain Here we use the facts that , , from (10) and (14), and , , from (10) and (13).
Claim 1. is decreasing and for all .
Suppose that there exists a such that on and . Now, let Firstly, differentiating , we get Secondly, differentiating , we get Hence, from (24), (25), and the continuity property of ODEs, we can draw for all . This means can be extended to infinity. This is a contradiction, so the claim is true.
Moreover, using (24) and (25) again, we have the following inequality for :
Putting (22) into (27) yields
Before completing the proof, we want the following technical lemma.