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

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

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.

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 $\mathrm{\Lambda }={(1-{\partial }_x^2)}^{\frac{1}{2}}$.

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

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 $a<b$, 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 ${\parallel u_0\parallel }_{H^s(R)}$.

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 $p\in C^1([0,T)\times R,R)$.

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. □

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

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 ${\int }_R|u|dx<\mathrm{\infty }$, then there exists a constant $K>0$ such that the solution of problem (3) satisfies ${\parallel u_x\parallel }_{L^{\mathrm{\infty }}}\le K$.

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 ${\parallel u_x^{\epsilon }\parallel }_{L^{\mathrm{\infty }}}\le 2{\int }_R|u|dx<K$. 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 ${\parallel u_x\parallel }_{L^{\mathrm{\infty }}}\le K$. □

Lemma 3.5 (Kato and Ponce [23])

where c is a constant depending only on r.

Lemma 3.6 (Kato and Ponce [23])

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

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

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 ${\parallel u_x\parallel }_{L^{\mathrm{\infty }}}=+\mathrm{\infty }$.

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 ${lim}_{t\to T}inf\{{inf}_{x\in R}u{u}_x(t,x)\}=\mathrm{\infty }$.