The global solution and blow-up phenomena to a modified Novikov equation

A modified Novikov equation with symmetric coefficients is investigated. Provided that the initial value u_0\in H^s(R) (s>\frac{3}{2}), (1-{\partial }_x^2)u_0 does not change sign and the solution u itself belongs to L^1(R), the existence and uniqueness of the global strong solutions to the equation are established in the space C([0,\mathrm{\infty });H^s(R))\cap C^1([0,\mathrm{\infty });H^{s-1}(R)). A blow-up result to the development of singularities in finite time for the equation is acquired.

MSC:35G25, 35L05.

Many scholars have paid attention to the integrable equation

which was derived by Novikov [1]. Well-posedness of the Novikov equation in the Sobolev spaces on the torus was first done by Tiglay in [2], and was completed on both the line and the circle by Himonas and Holliman in [3]. Its Hölder continuity properties were studied in Himonas and Holmes [4]. The periodic and the non-periodic Cauchy problem for Eq. (1) and continuity results for the data-to-solution map in the Sobolev spaces are discussed in Grayshan [5]. A matrix Lax pair for Eq. (1) is acquired in [6] and is shown to be related to a negative flow in the Sawada-Kotera hierarchy. The scattering theory is applied to find non-smooth explicit soliton solutions with multiple peaks for Eq. (1) in [7]. Sufficient conditions on the initial data to guarantee the formation of singularities in finite time for Eq. (1) are given in Jiang and Ni [8]. This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations [9–11]. Mi and Mu [12] established many dynamic results for a modified Novikov equation with peak solution. It is shown in Ni and Zhou [13] that the Novikov equation associated with initial value has locally well-posedness in a Sobolev space H^s with s>\frac{3}{2} by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for Eq. (1) are established in [13]. Using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, Yan et al. [14] proved the global existence and blow-up phenomena for the weakly dissipative Novikov equation. For other methods to handle the Novikov equation and the related partial differential equations, the reader is referred to [15–22] and the references therein.

Observing the coefficients of the Novikov equation (1), we see that the coefficient of u^2{u}_x is equal to the coefficient of u{u}_x{u}_{xx} plus the coefficient of u^2{u}_{xxx}. That is,

Indeed, this relationship among the coefficients plays important roles in the study of the essential dynamical properties of the Novikov model [1, 2, 11–13]. This motivates us to study the following equation:

where a>0 and b>0 are arbitrary constants. Clearly, letting a=3 and b=1, Eq. (2) becomes the Novikov equation (1). The essential difference between Eq. (2) and the Novikov equation (1) is that Eq. (2) does not conform with the following conservation law:

which results in the bounds of {\parallel u(t,\cdot )\parallel }_{L^{\mathrm{\infty }}(R)} for Eq. (1).

Making use of u_0\in H^s(R), s>\frac{3}{2}, the assumption that (1-{\partial }_x^2)u_0 does not change sign, and the assumption that the solution of Eq. (2) satisfies u\in L^1(R), we prove the global existence theorem of Eq. (2) in the Sobolev space,

The objective of this work is to investigate Eq. (2). Since a>0 and b>0 are arbitrary constants, we cannot obtain the boundedness of the solution u for Eq. (2) although the initial data satisfy the sign condition. To overcome this, assuming that the solution itself satisfies u\in L^1(R) and the initial data satisfy the sign condition, we adopt the methods used in Rodriguez-Blanco [16] to derive that {\parallel \frac{\partial u(t,x)}{\partial x}\parallel }_{L^{\mathrm{\infty }}(R)} possesses bounds for any time t>0. This leads us to establish the well-posedness of the global strong solutions to Eq. (2). Parts of the main results in [17, 18] are extended. In addition, we acquire a blow-up result to the development of singularities in finite time, which includes the blow-up result in [12].

The rest of this paper is organized as follows. Section 2 states the main results of this work. Section 3 proves the global existence result. The proof of a blow-up result is given in Section 4.

We let L^p=L^p(R) () be the space of all measurable functions h such that . We define L^{\mathrm{\infty }}=L^{\mathrm{\infty }}(R) with the standard norm {\parallel h\parallel }_{L^{\mathrm{\infty }}}={inf}_{m(e)=0}{sup}_{x\in R\mathrm{\setminus }e}|h(t,x)|. For any real number s, we let H^s=H^s(R) denote the Sobolev space with the norm defined by

where \hat{h}(t,\xi )={\int }_R{e}^{-ix\xi }h(t,x)dx. Here we note that the norms {\parallel \cdot \parallel }_{L^p}^p, {\parallel \cdot \parallel }_{L^{\mathrm{\infty }}} and {\parallel \cdot \parallel }_{H^s} depend on variable t.

For T>0 and nonnegative number s, C([0,T);H^s(R)) denotes the Frechet space of all continuous H^s-valued functions on [0,T). We set \mathrm{\Lambda }={(1-{\partial }_x^2)}^{\frac{1}{2}}.

In order to study the existence of solutions for Eq. (2), we consider its Cauchy problem in the form

which is equivalent to

where a>0 and b>0 are arbitrary constants. Now we give the main results for problem (3).

Theorem 1 Assume that the solution of problem (3) satisfies u(t,x)\in L^1(R) and let u_0(x)\in H^s, s>\frac{3}{2} and (1-{\partial }_x^2)u_0\ge 0 for all x\in R (or equivalently (1-{\partial }_x^2)u_0\le 0 for all x\in R). Then problem (3) has a unique solution satisfying

Theorem 2 Assume that u_0(x)\in H^s(R) with s>\frac{3}{2}. If a=b, then every solution of problem (3) exists globally in time. If a>b, then the solution blows up in finite time if and only if u{u}_x becomes unbounded from below in finite time. If , then the solution blows up in finite time if and only if u{u}_x becomes unbounded from above in finite time.

For proving the global existence for problem (3), we cite the local well-posedness result presented in [18].

Lemma 3.1 ([18])

Let u_0(x)\in H^s(R) with s>\frac{3}{2}. Then the Cauchy problem (3) has a unique solution u(t,x)\in C([0,T);H^s(R))\cap C^1([0,T);H^{s-1}(R)) where T>0 depends on {\parallel u_0\parallel }_{H^s(R)}.

Assume u_0\in H^s(R) with s>\frac{3}{2}. Then there exists a unique solution u(t,x) to problem (3) and

with the maximal existence time T>0. First, we study the differential equation

Lemma 3.2 Let u_0\in H^s, s>3 and let T>0 be the maximal existence time of the solution to problem (3). Then problem (5) has a unique solution p\in C^1([0,T)\times R,R). Moreover, the map p(t,\cdot ) is an increasing diffeomorphism of R with p_x(t,x)>0 for (t,x)\in [0,T)\times R.

Proof From Lemma 3.1, we have u\in C^1([0,T);H^{s-1}(R)) and H^{s-1}\in C^1(R). Thus we conclude that both functions u(t,x) and u_x(t,x) are bounded, Lipschitz in space and C^1 in time. Using the existence and uniqueness theorem of ordinary differential equations derives that problem (5) has a unique solution p\in C^1([0,T)\times R,R).

Differentiating Eq. (5) with respect to x yields

which leads to

For every , using the Sobolev imbedding theorem yields

It is inferred that there exists a constant K_0>0 such that p_x(t,x)\ge e^{-K_{0}t} for (t,x)\in [0,T)\times R. It completes the proof. □

Lemma 3.3 Let u_0\in H^s with s>3, and let T>0 be the maximal existence time of the problem (3). We have

where (t,x)\in [0,T)\times R and y:=u-u_{xx}.

Proof Using Eqs. (2) and (6)-(8), we have

Using p_x(0,x)=1 and solving the above equation, we complete the proof of the lemma. □

Remark 1 From Lemma 3.3, we conclude that, if u_0-u_{0xx}=(1-{\partial }_x^2)u_0\ge 0, then (1-{\partial }_x^2)u(t,x)\ge 0. Since the operator {(1-{\partial }_x^2)}^{-1} preserves positivity, we get u\ge 0. Similarly, if (1-{\partial }_x^2)u_0\le 0, we have (1-{\partial }_x^2)u\le 0 and u\le 0.

Lemma 3.4 If u_0\in H^s, s>\frac{3}{2}, such that (1-{\partial }_x^2)u_0\ge 0 (or (1-{\partial }_x^2)u_0\le 0) and , then there exists a constant K>0 such that the solution of problem (3) satisfies {\parallel u_x\parallel }_{L^{\mathrm{\infty }}}\le K.

Proof We will prove this lemma to assume u_0\in H^{\mathrm{\infty }} which results in u\in H^{\mathrm{\infty }} from Lemma 3.1. For (1-{\partial }_x^2)u_0\ge 0, from Lemma 3.3, we have (1-{\partial }_x^2)u\ge 0. Then u\ge 0 does not change sign. From the assumption one derives

where c is a positive constant. Then

On the other hand, we have

which results in

We conclude from Eqs. (11) and (13) that {\parallel u_x\parallel }_{L^{\mathrm{\infty }}}\le K. To complete the proof, we use a simple density argument [16]. Setting u_0^{\epsilon }=e^{\epsilon {\partial }_x^2}{u}_0, we have u_0^{\epsilon }\in H^{\mathrm{\infty }} and . Applying {\parallel u_x^{\epsilon }-u_x\parallel }_{L^{\mathrm{\infty }}}\le {sup}_{[0,T]}{\parallel u_x^{\epsilon }-u_x\parallel }_{H^s}\to 0 when \epsilon \to 0, we have {\parallel u_x\parallel }_{L^{\mathrm{\infty }}}\le K. □

Using the first equation of system (3) one derives

from which we have the conservation law

Lemma 3.5 (Kato and Ponce [23])

If r\ge 0, then H^r\cap L^{\mathrm{\infty }} is an algebra. Moreover

where c is a constant depending only on r.

Lemma 3.6 (Kato and Ponce [23])

Let r>0. If u\in H^r\cap W^{1,\mathrm{\infty }} and v\in H^{r-1}\cap L^{\mathrm{\infty }}, then

Lemma 3.7 Let s>\frac{3}{2} and the function u(t,x) is a solution of problem (3) and the initial data u_0(x)\in H^s(R). Then the following results hold:

For q\in (0,s-1], there is a constant c only depending on a and b such that

Proof Using |2u{u}_x|\le (u^2+u_x^2), the Gronwall inequality and Eq. (14), one derives Eq. (15).

Using {\partial }_x^2=-{\mathrm{\Lambda }}^2+1 and the Parseval equality gives rise to

For q\in (0,s-1], applying ({\mathrm{\Lambda }}^{q}u){\mathrm{\Lambda }}^q to both sides of the first equation of system (3) and integrating with respect to x by parts, we have the identity

We will estimate the terms on the right-hand side of Eq. (17) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.5 and 3.6, we have

Using the above estimate to the second term yields

Using the Cauchy-Schwartz inequality and Lemma 3.5, we obtain

For the last term in Eq. (17), using u{(u_x^2)}_x={(u{u}_x^2)}_x-u_x{u}_x^2 results in

For K_1, it follows from Eq. (20) that

For K_2, applying Lemma 3.5 derives

It follows from Eqs. (18)-(23) that there exists a constant c such that

Integrating both sides of the above inequality with respect to t results in inequality (16). □

Proof of Theorem 1 Using Eq. (16) with q=s-1, we obtain

Applying the Gronwall inequality, we get

Using Eq. (15) and Lemma 3.4, we complete the proof of Theorem 1. □

Remark 2 In fact, using {\parallel u_x\parallel }_{L^{\mathrm{\infty }}}\le {\parallel u\parallel }_{H^s} with s>\frac{3}{2}, Eqs. (15) and (26), we derive that the solution of Eq. (2) in space H^s(R) blows up in finite time if and only if {\parallel u_x\parallel }_{L^{\mathrm{\infty }}}=+\mathrm{\infty }.

Multiplying Eq. (2) by y=u-u_{xx} and integrating by parts, we get

When a=b, from Eq. (27), we derive {\parallel u_x\parallel }_{L^{\mathrm{\infty }}} is bounded. From Lemma 3.7 and Remark 2, we see that problem (3) has a global solution in the space

Assume that the solution u=u(\cdot ,u_0) of problem (3) blows up in finite time in the space H^s(R) with s>\frac{3}{2}. If , we assume that u{u}_x is bounded from below on [0,T)\times R, i.e., there exists a constant M>1 such that

From Eq. (27), we get

from which we derive that the H^2 norm of the solution to problem (3) does not blow up in finite time. From Remark 2, we know that this is impossible. Therefore, we have {lim}_{t\to T}inf\{{inf}_{x\in R}u{u}_x(t,x)\}=-\mathrm{\infty }.

Similar to the above, we know that if b-a>0, the solution of problem (3) blows up if and only if {lim}_{t\to T}inf\{{inf}_{x\in R}u{u}_x(t,x)\}=\mathrm{\infty }.

This work is supported by both the Fundamental Research Funds for the Central Universities (JBK130401, JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).

Affiliations

1. Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

   • Shaoyong Lai
   • , Haibo Yan
   •  & Nan Li

Corresponding author

Correspondence to Shaoyong Lai.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The article is a joint work of three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

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