Skip to main content

A necessary and sufficient condition of blow-up for a nonlinear equation


We investigate a nonlinear equation with quadratic nonlinearities, including a nonlinear model in Silva and Freire (J. Differ. Equ. 320:371–398, 2022). Using the classical energy estimate methods, we give a necessary and sufficient condition of blow-up of solutions to nonlinear equations. We answer a problem pointed out by Silva and Freire (J. Differ. Equ. 320:371–398, 2022).

1 Introduction

Silva and Freire [1] investigated in detail the following equation:

$$\begin{aligned} W_{t}-W_{txx}=-WW_{x}+WW_{xxx}, \quad (t,x)\in \mathbb{R}\mathbbm{_{+}} \times \mathbb{R}\mathbbm{,} \end{aligned}$$

for which they considered continuation and persistence of solutions and necessary conditions for blow-up of a solution.

Equation (1.1) is related to the equation

$$\begin{aligned} W_{t}-W_{txx}+aW^{k}W_{x}=bW^{k-1}W_{x}W_{xx}+cW^{k}W_{xxx}, \end{aligned}$$

where constants \(a,b,c\) satisfy \((ab,ac)\neq (0,0)\), and \(k\neq 0\) (see [2]). Under certain restrictions on the parameters \(a,b,c\), and k, the conserved currents, peakon solutions, and point symmetries are discussed in [24]. Obviously, when \(a=3, b=2,c=1\), and \(k=1\), Eq. (1.2) reduces to the standard Camassa–Holm equation [5]. If \(a=4, b=3,c=1\), and \(k=1\), then Eq. (1.2) becomes the Degasperis–Procesi model [6]. When \(a=4, b=3, c=1\), and \(k=2\), Eq. (1.2) reduces to the Novikov equation [7]. For \(a=b+c\), \(b\in \mathbb{R}\), \(c\neq 0\), and \(k>0\), if the initial value belongs to a suitable Besov space, the well-posedness of short-time solutions for Eq. (1.2) is investigated in [8]. Under certain restrictions on the constants \(a,b,c,k\), the global well-posedness for Eq. (1.2) is also established in Yan [8]. For real b, \(c=1\), and \(a=b+1\), the traveling wave solutions, the persistence properties, and unique continuation to Eq. (1.2) are considered by Guo et al. [9, 10] and Himonas and Thompson [11, 12]. Under different assumptions on the parameters \(a,b,c,k\) and the initial data, many useful dynamical properties for Eq. (1.2) can be found in [1317].

We consider the following initial value problem:

$$\begin{aligned} \textstyle\begin{cases} W_{t}-W_{txx}=-mWW_{x}+WW_{xxx}, \\ W(0,x)=W_{0}(x), \end{cases}\displaystyle \end{aligned}$$

where the constant \(m\in (-\infty , \infty )\). If \(m=1\), then the first equation in (1.3) becomes Eq. (1.1).

For problem (1.3) with \(m=1\), Silva and Freire [1] pointed out the following conjecture.


Let \(m=1, s>\frac{3}{2}, W_{0}(x)\in H^{s}(\mathbb{R})\), and lifespan \(T>0\). Then the solution \(W(t,x)\) of problem (1.3) blows up at finite time if and only if

$$\begin{aligned} \lim_{t\rightarrow T} \bigl\Vert W_{x}(t,\cdot ) \bigr\Vert _{L^{\infty}}= \infty . \end{aligned}$$

The conjecture is presented on p. 396 in [1]. We will derive several estimates from problem (1.3) itself. Using the obtained estimates, we obtain two results: (1) If \(W_{0}(x)\in H^{s}(\mathbb{R}), s>\frac{3}{2}\), and the solution of problem (1.3) blows up, then \(\int _{0}^{T}|W_{x}(t,x)|\,dx=\infty \), where T is the lifespan of \(W(t,x)\) (2) If \(W_{0}(x)\in H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\), then \(\lim_{t\rightarrow T}\Vert W(t,\cdot )\Vert _{H^{s}}= \infty \) if and only if (1.4) holds. Our Theorem 3.2 demonstrates that the conjecture is right for any constant \(m\in (-\infty , \infty )\).

In Sect. 2, we present several lemmas, and in Sect. 3, we provide our main results and their proofs.

2 Several lemmas

Set \(\Lambda ^{2}=1-\partial _{x}^{2}\). Then \(\partial _{x}^{2}=1-\Lambda ^{2}\) and \(\Lambda ^{-2}=(1-\partial _{x}^{2})^{-1}\), and we have

$$\begin{aligned} W_{t}&=\Lambda ^{-2}(WW_{xxx})-m\Lambda ^{-2}(WW_{x}) \\ & =\Lambda ^{-2} \bigl((WW_{xx})_{x}-W_{x}W_{xx} \bigr)-m \Lambda ^{-2}(WW_{x}) \\ & =\Lambda ^{-2} \bigl( \bigl((WW_{x})_{x}-W_{x}^{2} \bigr)_{x}-W_{x}W_{xx} \bigr)-m\Lambda ^{-2}(WW_{x}) \\ & =\Lambda ^{-2} \bigl((WW_{x})_{xx}-3W_{x}W_{xx} \bigr)-m \Lambda ^{-2}(WW_{x}) \\ & =\Lambda ^{-2} \bigl(1-\Lambda ^{2} \bigr) (WW_{x})-3\Lambda ^{-2}(W_{x}W_{xx})-m \Lambda ^{-2}(WW_{x}) \\ & =-WW_{x}-3\Lambda ^{-2}(W_{x}W_{xx})+ \frac{1-m}{2} \Lambda ^{-2} \bigl(W^{2} \bigr)_{x}. \end{aligned}$$

Thus problem (1.3) becomes

$$\begin{aligned} \textstyle\begin{cases} W_{t}+WW_{x}=-3\Lambda ^{-2}(W_{x}W_{xx})+\frac{1-m}{2}\Lambda ^{-2}(W^{2})_{x}, \\ W(0,x)=W_{0}(x). \end{cases}\displaystyle \end{aligned}$$

Lemma 2.1

Let \(W_{0}\in H^{s}(\mathbb{R}) \) with \(s>\frac{3}{2}\). Then there is \(T=T(W_{0})>0\) such that problem (2.1) has a unique solution \(W(t,x)\), and

$$\begin{aligned} W\in C \bigl([0,T);H^{s}(\mathbb{R})\bigr)\cap C^{1} \bigl([0,T);H^{s-1}(\mathbb{R})\bigr). \end{aligned}$$

Using the Kato theorem [18], we can prove the well-posedness of local solutions for problem (2.1). In fact, the proof of well-posedness of a short-time solution for problem (2.1) is very similar to those of the famous Camassa–Holm and Degasperis–Procesi models (see [11, 15, 16]). Here we omit its proof.

Lemma 2.2

Suppose that \(s\geq 3\) and \(W(t,x)\in H^{s}(\mathbb{R})\). Then

$$\begin{aligned} &\int _{\mathbb{R}}WW_{x}W_{xx}\,dx=- \frac{1}{2} \int _{\mathbb{R}}W_{x}^{3}\,dx, \end{aligned}$$
$$\begin{aligned} &\int _{\mathbb{R}}WW_{xx}W_{xxx}\,dx=- \frac{1}{2} \int _{\mathbb{R}}W_{x}W_{xx}^{2}\,dx. \end{aligned}$$


SinceFootnote 1

$$\begin{aligned} \int _{\mathbb{R}}WW_{x}W_{xx}\,dx&= \int _{\mathbb{R}}WW_{x}\,dW_{x} \\ & = \bigl(WW_{x}^{2} \bigr)\big|_{-\infty}^{\infty}- \int _{ \mathbb{R}}W_{x} \bigl(W_{x}^{2}+WW_{xx} \bigr)\,dx, \\ & =- \int _{\mathbb{R}}W_{x} \bigl(W_{x}^{2}+WW_{xx} \bigr)\,dx, \end{aligned}$$

we get (2.2). Similarly, we have

$$\begin{aligned} \int _{\mathbb{R}}WW_{xx}W_{xxx}\,dx&= \int _{\mathbb{R}}WW_{xx}\,dW_{xx} \\ & = \bigl(WW_{xx}^{2} \bigr)\big|_{-\infty}^{\infty}- \int _{\mathbb{R}}W_{xx}(W_{x}W_{xx}+WW_{xxx}) \,dx, \\ & = - \int _{\mathbb{R}}W_{xx}(W_{x}W_{xx}+WW_{xxx}) \,dx, \end{aligned}$$

which leads to (2.3). □

Lemma 2.3

Let \(W_{0}(x)\in H^{s}(\mathbb{R})\) \((s>\frac{3}{2})\). Then

$$\begin{aligned} \int _{\mathbb{R}}\Lambda ^{-2} \bigl(W^{2} \bigr) \,dx= \int _{\mathbb{R}}W^{2}\,dx,\qquad \int _{\mathbb{R}}\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)\,dx= \int _{ \mathbb{R}}W_{x}^{2}\,dx. \end{aligned}$$


We only need to prove the first identity in (2.4). Since

$$\begin{aligned} \Lambda ^{-2}W^{2}=\frac{1}{2} \int _{\mathbb{R}}e^{-|x-\eta |} W^{2}(t, \eta )\,d\eta \geq 0 \end{aligned}$$


$$\begin{aligned} \int _{\mathbb{R}}e^{-|x-\eta |}\,d\eta =2, \end{aligned}$$

by the Tonelli theorem we get

$$\begin{aligned} \int _{\mathbb{R}}\Lambda ^{-2} \bigl(W^{2} \bigr) \,dx&=\frac{1}{2} \int _{ \mathbb{R}} \int _{\mathbb{R}}e^{-|x-\eta |}W^{2}(t,\eta )\,d\eta \,dx \\ & =\frac{1}{2} \int _{\mathbb{R}}W^{2}(t, \eta )\,d\eta \int _{\mathbb{R}}e^{-|x-\eta |}\,dx \\ & = \int _{\mathbb{R}}W^{2}(t,\eta )\,d \eta , \end{aligned}$$

which finishes the proof. □

Lemma 2.4


If \(r\geq 0\) and \(f_{1},f_{2}\in H^{r}\cap L^{\infty}\), then

$$\begin{aligned} \Vert f_{1}f_{2} \Vert _{r}\leq c \bigl( \Vert f_{1} \Vert _{L^{\infty}} \Vert f_{2} \Vert _{r}+ \Vert f_{1} \Vert _{r} \Vert f_{2} \Vert _{L^{\infty}} \bigr), \end{aligned}$$

where the constant \(c>0\) depends only on r.

Lemma 2.5


Let \(f_{1}\in H^{r}\cap W^{1,\infty}\) \((r>0)\) and \(f_{2}\in H^{r-1}\cap L^{\infty}\). Then

$$\begin{aligned} \bigl\Vert \bigl[\Lambda ^{r},f_{1} \bigr]f_{2} \bigr\Vert _{L^{2}}\leq c \bigl( \Vert \partial _{x}f_{1} \Vert _{L^{\infty}} \bigl\Vert \Lambda ^{r-1}f_{2} \bigr\Vert _{L^{2}}+ \bigl\Vert \Lambda ^{r}f_{1} \bigr\Vert _{L^{2}} \Vert f_{2} \Vert _{L^{\infty}} \bigr), \end{aligned}$$

where \([\Lambda ^{r},f_{1} ]=\Lambda ^{r}f_{1}-f_{1}\Lambda ^{r}\), and the constant \(c>0\) depends only on r.

Remark 1

Using the arguments in [8, 15], the lifespan T in Lemma 2.1 does not depend on the Sobolev index \(s>\frac{3}{2}\). Namely, for arbitrary \(s_{1}>s>\frac{3}{2}\) or \(s>s_{1}>\frac{3}{2}\), the maximal existence time for \(\Vert W\Vert _{H^{s}}\) and \(\Vert W\Vert _{H^{s_{1}}}\) is the same.

3 Main results

Theorem 3.1

Let \(W_{0}\in H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\), and suppose W satisfies problem (1.3) or problem (2.1). If the lifespan T of W is finite and

$$\begin{aligned} \lim_{t\rightarrow T} \bigl\Vert W(t,\cdot ) \bigr\Vert _{H^{s}}= \infty , \end{aligned}$$


$$\begin{aligned} \int _{0}^{T} \bigl\Vert W_{x}(\tau , \cdot ) \bigr\Vert _{L^{\infty}}\,d \tau =\infty . \end{aligned}$$


If \(s>\frac{3}{2}\), then using the operator \(\Lambda ^{s}W\Lambda ^{s}\), from problem (2.1) we obtain

$$\begin{aligned} & \frac{1}{2}\frac{d}{dt} \int _{ \mathbb{R}} \bigl(\Lambda ^{s}W \bigr)^{2} \,dx \\ &\quad= \int _{ \mathbb{R}} \bigl(\Lambda ^{s}W \bigr)\Lambda ^{s}W_{t}\,dx \\ &\quad= \int _{\mathbb{R}} \bigl(\Lambda ^{s}W \bigr)\Lambda ^{s} \biggl(-WW_{x}- \frac{3}{2}\Lambda ^{-2} \partial _{x} \bigl(W_{x}^{2} \bigr)+ \frac{1-m}{2} \Lambda ^{-2} \bigl(W^{2} \bigr)_{x} \biggr)\,dx, \end{aligned}$$

which leads to

$$\begin{aligned} & \frac{1}{2}\frac{d}{dt} \int _{ \mathbb{R}} \bigl(\Lambda ^{s}W \bigr)^{2} \,dx \\ &\quad\leq \biggl\vert \int _{\mathbb{R}} \bigl(\Lambda ^{s}W \bigr)\Lambda ^{s}(WW_{x})\,dx \biggr\vert + \frac{ \vert m-1 \vert }{2} \biggl\vert \int _{\mathbb{R}} \bigl(\Lambda ^{s}W \bigr)\Lambda ^{s-2} \bigl(W^{2} \bigr)_{x}\,dx \biggr\vert \\ &\qquad{}+\frac{3}{2} \biggl\vert \int _{\mathbb{R}}\Lambda ^{s}W \Lambda ^{s-2} \partial _{x} \bigl(W_{x}^{2} \bigr)\,dx \biggr\vert \\ &\quad= G_{1}+G_{2}+G_{3}. \end{aligned}$$

In fact, we have

$$\begin{aligned} \int _{\mathbb{R}}W\Lambda ^{s}W\Lambda ^{s}W_{x} \,dx&= \int _{ \mathbb{R}} W\Lambda ^{s}W\,d \bigl(\Lambda ^{s}W \bigr) \\ & =- \int _{\mathbb{R}}\Lambda ^{s}W \bigl(W_{x}\Lambda ^{s}W+W \Lambda ^{s}W_{x} \bigr)\,dx, \end{aligned}$$

from which we obtain

$$\begin{aligned} \int _{\mathbb{R}} W\Lambda ^{s}W\Lambda ^{s}W_{x} \,dx=-\frac{1}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{s}W\Lambda ^{s}W\,dx. \end{aligned}$$

Employing the Cauchy–Schwarz inequality, (3.4), and Lemma 2.5, we acquire

$$\begin{aligned} \biggl\vert \int _{ \mathbb{R}} \bigl(\Lambda ^{s}W \bigr)\Lambda ^{s}(WW_{x})\,dx \biggr\vert ={}& \biggl\vert \int _{\mathbb{R}} \bigl(\Lambda ^{s}W \bigr) \bigl(\Lambda ^{s}(WW_{x})-W \Lambda ^{s}W_{x} \bigr) \,dx \\ &{}+ \int _{ \mathbb{R}} \bigl(\Lambda ^{s}W \bigr)W\Lambda ^{s}W_{x}\,dx \biggr\vert \\ \leq{}& \biggl\vert \int _{\mathbb{R}} \bigl(\Lambda ^{s}W \bigr) \bigl( \Lambda ^{s}(WW_{x})-W\Lambda ^{s}W_{x} \bigr) \,dx \biggr\vert \\ &{}+ \biggl\vert \int _{\mathbb{R}} \bigl(\Lambda ^{s}W \bigr)W\Lambda ^{s}W_{x}\,dx \biggr\vert \\ \leq{}& c \Vert W \Vert _{H^{s}} \bigl( \Vert W \Vert _{H^{s-1}} \Vert W_{x} \Vert _{L^{\infty}}+ \Vert W \Vert _{H^{s}} \Vert W_{x} \Vert _{L^{\infty}} \bigr) \\ &{}+\frac{1}{2} \Vert W_{x} \Vert _{L^{ \infty}} \bigl\Vert \Lambda ^{s}W \bigr\Vert _{L^{2}} \\ \leq{}& c \Vert W_{x} \Vert _{L^{\infty}} \Vert W \Vert _{H^{s}}^{2}, \end{aligned}$$

which leads to

$$\begin{aligned} G_{1}\leq c \Vert W_{x} \Vert _{L^{\infty}} \Vert W \Vert _{H^{s}}^{2}. \end{aligned}$$

Similarly to the proof of (3.5), we have

$$\begin{aligned} G_{2}&\leq \frac{ \vert m-1 \vert }{2} \biggl\vert \int _{\mathbb{R}} \bigl(\Lambda ^{s-1}W \bigr) \Lambda ^{s-1} \bigl(W^{2} \bigr)_{x}\,dx \biggr\vert \\ & \leq c \biggl\vert \int _{\mathbb{R}} \bigl(\Lambda ^{s-1}W \bigr)\Lambda ^{s-1}(WW_{x})\,dx \biggr\vert \\ & \leq c \Vert W_{x} \Vert _{L^{\infty}} \Vert W \Vert _{H^{s-1}}^{2} \\ & \leq c \Vert W_{x} \Vert _{L^{\infty}} \Vert W \Vert _{H^{s}}^{2}. \end{aligned}$$

Now Lemma 2.4 yields

$$\begin{aligned} G_{3}&\leq \bigl\Vert \Lambda ^{s}W \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{s-2}\partial _{x} \bigl(W_{x}^{2} \bigr) \bigr\Vert _{L^{2}} \\ & \leq c \bigl\Vert \Lambda ^{s}W \bigr\Vert _{L^{2}} \bigl\Vert W_{x}^{2} \bigr\Vert _{H^{s-1}} \\ & \leq c \bigl\Vert \Lambda ^{s}W \bigr\Vert _{L^{2}} \Vert W_{x} \Vert _{L^{\infty}} \Vert W_{x} \Vert _{H^{s-1}} \\ & \leq c \Vert W_{x} \Vert _{L^{\infty}} \Vert W \Vert _{H^{s}}^{2}. \end{aligned}$$

Using inequalities (3.3), (3.5),(3.6), and (3.7) results in

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{-\infty}^{\infty} \bigl(\Lambda ^{s}W \bigr)^{2}\,dx \leq c \Vert W_{x} \Vert _{L^{\infty}} \bigl\Vert \Lambda ^{s}W \bigr\Vert _{L^{2}}^{2}, \end{aligned}$$

where \(c>0\) is a constant. Using (3.8) yields

$$\begin{aligned} \Vert W \Vert _{H^{s}}\leq \Vert W_{0} \Vert _{H^{s}}e^{c \int _{0}^{t} \Vert W_{x} \Vert _{L^{\infty}} \,d\tau}. \end{aligned}$$

Suppose that \(\lim_{t\rightarrow T}\Vert W\Vert _{H^{s}}=\infty \). From (3.9) we have

$$\begin{aligned} \int _{0}^{T} \Vert W_{x} \Vert _{L^{\infty}}\,d\tau =\infty , \end{aligned}$$

which ends the proof. □

Theorem 3.2

Let \(W_{0}(x)\in H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\), and let T be the lifespan of solution \(W(t,x)\) for problem (2.1). If T is finite, then

$$\begin{aligned} \lim_{t\rightarrow T} \bigl\Vert W(t,\cdot ) \bigr\Vert _{H^{s}( \mathbb{R})}=\infty \end{aligned}$$

if and only if

$$\begin{aligned} \lim_{t\rightarrow T} \bigl\Vert W_{x}(t,\cdot ) \bigr\Vert _{L^{ \infty}(\mathbb{R})}=\infty . \end{aligned}$$


Let (3.10) hold. We will derive that (3.11) holds. Using Remark 1 and choosing \(s=3\), Lemma 2.1 ensures that there exists \(W(t,x)\in C([0,T), H^{3}(\mathbb{R}))\cap C^{1}([0,T), H^{2}( \mathbb{R}))\). We will employ the classical energy estimates. From problem (2.1) we acquire

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{\mathbb{R}}W^{2}\,dx&= \int _{\mathbb{R}}WW_{t}\,dx \\ & = \int _{\mathbb{R}} W \bigl(-WW_{x}-3\Lambda ^{-2}(W_{x}W_{xx}) \bigr)\,dx+\frac{1-m}{2} \int _{\mathbb{R}}W\Lambda ^{-2} \bigl(W^{2} \bigr)_{x}\,dx \\ & =-3 \int _{\mathbb{R}}W\Lambda ^{-2}(W_{x}W_{xx}) \,dx+ \frac{1-m}{2} \int _{\mathbb{R}}W\Lambda ^{-2} \bigl(W^{2} \bigr)_{x}\,dx \\ & =-\frac{3}{2} \int _{\mathbb{R}}W\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)_{x}\,dx+ \frac{1-m}{2} \int _{\mathbb{R}}W\Lambda ^{-2} \bigl(W^{2} \bigr)_{x}\,dx \\ & =\frac{3}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)\,dx- \frac{1-m}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2} \bigl(W^{2} \bigr)\,dx. \end{aligned}$$

Applying the first equation in (2.1) yields

$$\begin{aligned} W_{tx}={}&{-}W_{x}^{2}-WW_{xx}- \frac{3}{2}\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)_{xx}+ \frac{1-m}{2}\Lambda ^{-2} \bigl(W^{2} \bigr)_{xx} \\ ={}&{-}W_{x}^{2}-WW_{xx}-\frac{3}{2}\Lambda ^{-2} \bigl(1-\Lambda ^{2} \bigr) \bigl(W_{x}^{2} \bigr) \\ &{}+\frac{1-m}{2}\Lambda ^{-2} \bigl(1- \Lambda ^{2} \bigr) \bigl(W^{2} \bigr) \\ ={}&{-}W_{x}^{2}-WW_{xx}-\frac{3}{2}\Lambda ^{-2}W_{x}^{2}+ \frac{3}{2}W_{x}^{2} \\ &{}+\frac{1-m}{2}\Lambda ^{-2} \bigl(W^{2} \bigr)- \frac{1-m}{2}W^{2} \\ ={}&\frac{1}{2}W_{x}^{2}-WW_{xx}- \frac{1-m}{2}W^{2}- \frac{3}{2}\Lambda ^{-2}W_{x}^{2}+ \frac{1-m}{2}\Lambda ^{-2} \bigl(W^{2} \bigr). \end{aligned}$$

Using Lemma 2.2 and (3.13), we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{\mathbb{R}}W_{x}^{2}\,dx={}& \int _{ \mathbb{R}}W_{x} \biggl(\frac{1}{2}W_{x}^{2}-WW_{xx}- \frac{1-m}{2}W^{2} - \frac{3}{2}\Lambda ^{-2}W_{x}^{2} \\ &{}+\frac{1-m}{2}\Lambda ^{-2}W^{2} \biggr)\,dx \\ ={}&\frac{1}{2} \int _{\mathbb{R}}W_{x}^{3}\,dx- \int _{ \mathbb{R}}WW_{x}W_{xx}\,dx- \frac{3}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2}W_{x}^{2} \,dx \\ &{}+\frac{1-m}{2} \int _{ \mathbb{R}}W_{x}\Lambda ^{-2}W^{2} \,dx \\ ={}& \int _{\mathbb{R}}W_{x}^{3}\,dx-\frac{3}{2} \int _{ \mathbb{R}}W_{x}\Lambda ^{-2}W_{x}^{2} \,dx+ \frac{1-m}{2} \int _{ \mathbb{R}}W_{x}\Lambda ^{-2}W^{2} \,dx. \end{aligned}$$

Using (3.13) gives rise to

$$\begin{aligned} W_{txx}={}&W_{x}W_{xx}-W_{x}W_{xx}-WW_{xxx}-(1-m)WW_{x} \\ &{-}\frac{3}{2}\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)_{x}+ \frac{1-m}{2}\Lambda ^{-2} \bigl(W^{2} \bigr)_{x} \\ ={}&{-}WW_{xxx}-(1-m)WW_{x}-\frac{3}{2}\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)_{x}+ \frac{1-m}{2}\Lambda ^{-2} \bigl(W^{2} \bigr)_{x}. \end{aligned}$$

Applying integration by parts, (3.15), and Lemma 2.2, we have

$$\begin{aligned} & \frac{1}{2}\frac{d}{dt} \int _{\mathbb{R}}W_{xx}^{2}\,dx \\ &\quad=- \int _{\mathbb{R}}WW_{xx}W_{xxx}\,dx-(1-m) \int _{\mathbb{R}}WW_{x}W_{xx}\,dx \\ &\qquad{}-\frac{3}{2} \int _{\mathbb{R}}W_{xx} \Lambda ^{-2} \bigl(W_{x}^{2} \bigr)_{x}\,dx+ \frac{1-m}{2} \int _{\mathbb{R}}W_{xx} \Lambda ^{-2} \bigl(W^{2} \bigr)_{x}\,dx \\ &\quad=\frac{1}{2} \int _{\mathbb{R}}W_{x}W_{xx}^{2}\,dx+ \frac{1-m}{2} \int _{\mathbb{R}}W_{x}^{3}\,dx \\ &\qquad{}-\frac{3}{2} \int _{\mathbb{R}}W\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)_{xxx}\,dx -\frac{1-m}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2} \bigl(W^{2} \bigr)_{xx}\,dx \\ &\quad=\frac{1}{2} \int _{\mathbb{R}}W_{x}W_{xx}^{2}\,dx+ \frac{1-m}{2} \int _{\mathbb{R}}W_{x}^{3}\,dx - \frac{3}{2} \int _{\mathbb{R}}W\Lambda ^{-2} \bigl(1- \Lambda ^{2} \bigr) \bigl(W_{x}^{2} \bigr)_{x} \,dx \\ &\qquad{}-\frac{1-m}{2} \int _{\mathbb{R}}W_{x} \Lambda ^{-2} \bigl(1- \Lambda ^{2} \bigr) \bigl(W^{2} \bigr)\,dx \\ &\quad=\frac{1}{2} \int _{\mathbb{R}}W_{x}W_{xx}^{2}\,dx+ \frac{1-m}{2} \int _{\mathbb{R}}W_{x}^{3}\,dx+3 \int _{\mathbb{R}}WW_{x}W_{xx}\,dx \\ &\qquad{}-\frac{3}{2} \int _{\mathbb{R}}W\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)_{x}\,dx- \frac{1-m}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2}W^{2} \,dx \\ &\quad=\frac{1}{2} \int _{\mathbb{R}}W_{x}W_{xx}^{2}\,dx- \frac{m+2}{2} \int _{\mathbb{R}}W_{x}^{3}\,dx \\ &\qquad{}+\frac{3}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)\,dx -\frac{1-m}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2}W^{2} \,dx. \end{aligned}$$

Using (3.12), (3.14), and (3.16), we have

$$\begin{aligned} & \frac{1}{2}\frac{d}{dt} \int _{\mathbb{R}} \bigl(W^{2}+W_{x}^{2}+W^{2}_{xx} \bigr)\,dx \\ &\quad=-\frac{m}{2} \int _{\mathbb{R}}W_{x}^{3}\,dx + \frac{1}{2} \int _{ \mathbb{R}}W_{x}W_{xx}^{2}\,dx \\ &\qquad{}+\frac{m-1}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2}W^{2} \,dx +\frac{3}{2} \int _{\mathbb{R}}W_{x}\Lambda ^{-2} \bigl(W_{x}^{2} \bigr)\,dx. \end{aligned}$$

If (3.10) holds, then suppose that we can choose a positive constant M satisfying

$$\begin{aligned} \bigl\vert W_{x}(t,x) \bigr\vert < M,\quad t\in [0,T), x\in \mathbb{R}. \end{aligned}$$

Employing (3.17), (3.18), Lemma 2.3, \(\Lambda ^{-2}(W^{2})\geq 0\), and \(\Lambda ^{-2}(W_{x})^{2}\geq 0\), we have

$$\begin{aligned} & \frac{1}{2} \biggl[\frac{d}{dt} \int _{\mathbb{R}} \bigl(W^{2}+W_{x}^{2}+W_{xx}^{2} \bigr)\,dx \biggr] \\ &\quad< \frac{M \vert m \vert }{2} \int _{\mathbb{R}}W_{x}^{2}\,dx+\frac{M}{2} \int _{ \mathbb{R}}W_{xx}^{2}\,dx+\frac{ \vert m-1 \vert M}{2} \int _{\mathbb{R}}W^{2}\,dx+ \frac{3M}{2} \int _{\mathbb{R}}W_{x}^{2}\,dx \\ &\quad< \max \biggl\{ \frac{M \vert m \vert }{2},\frac{3M}{2},\frac{ \vert m-1 \vert M}{2} \biggr\} \int _{\mathbb{R}} \bigl(W^{2}+W_{x}^{2}+W_{xx}^{2} \bigr)\,dx. \end{aligned}$$


$$\begin{aligned} H(t)= \int _{\mathbb{R}} \bigl(W^{2}+W_{x}^{2}+W_{xx}^{2} \bigr)\,dx,\qquad K=\max \bigl\{ M \vert m \vert ,3M, \vert m-1 \vert M \bigr\} . \end{aligned}$$

From (3.19) we obtain

$$\begin{aligned} H(t)\leq H(0)+K \int _{0}^{t} H(\tau )\,d\tau , \end{aligned}$$

which, together with the Gronwall inequality, yields

$$\begin{aligned} H(t)\leq H(0)e^{Kt}. \end{aligned}$$

From (3.20) we obtain \(W(t,x)\in H^{2}(\mathbb{R})\), which, combined with Remark 1, is a contradiction to (3.10). Therefore we conclude that assumption (3.18) is not right.

Conversely, using \(\Vert W_{x}\Vert _{L^{\infty}}< c \Vert W\Vert _{H^{s}}\), if

$$\begin{aligned} \lim_{t\rightarrow T} \bigl\Vert W_{x}(t,\cdot ) \bigr\Vert _{L^{ \infty}(\mathbb{R})}=\infty , \end{aligned}$$

then we derive that

$$\begin{aligned} \lim_{t\rightarrow T} \bigl\Vert W(t,\cdot ) \bigr\Vert _{H^{s}}= \infty . \end{aligned}$$

The proof is completed. □

Availability of data and materials

Not applicable.


  1. For any \(f\in L^{r}(\mathbb{R})\) with \(1\leq r\leq \infty \), we have \(\Lambda ^{-2}f(x)=\frac{1}{2}\int _{\mathbb{R}}e^{-|x-\eta |}f(\eta )\,d \eta \) (see Constantin and Escher [14]). If a function \(g\in H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\), then \(g(\pm \infty )=g'(\pm \infty )=g''(\pm \infty )=g^{[s]}(\pm \infty )=0\), where \([s]\) denotes the integer part of s (see [18]).


  1. Silva, P.L., Freire, I.L.: Existence, persistence, and continuation of solutions for a generalized 0-Holm–Staley equation. J. Differ. Equ. 320, 371–398 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anco, S.C., Silva, P.L., Freire, I.L.: A family of wave-breaking equations generalizing the Camassa–Holm and Novikov equations. J. Math. Phys. 56, 091506 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Freire, I.L.: A look on some results about Camassa–Holm type equations. Commun. Math. 29, 115–130 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Freire, I.L.: Conserved quantities, continuation and compactly supported solutions of some shallow water models. J. Phys. A, Math. Theor. 54, 015207 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory, vol. 1, pp. 23–37. World Scientific, Singapore (1999)

    Google Scholar 

  7. Novikov, V.: Generalizations of the Camassa–Holm equation. J. Phys. A 42, 342002 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yan, K.: Wave-breaking and global existence for a family of peakon equations with high order nonlinearity. Nonlinear Anal., Real World Appl. 45, 721–735 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guo, Z.G., Li, K., Xu, C.: On generalized Camassa–Holm type equation with \((k+1)\)-degree nonlinearities. Z. Angew. Math. Mech. 98, 1567–1573 (2018)

    Article  MathSciNet  Google Scholar 

  10. Guo, Z.G., Li, X.G., Yu, C.: Some properties of solutions to the Camassa–Holm-type equation with higher order nonlinearities. J. Nonlinear Sci. 28, 1901–1914 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Himonas, A.A., Holliman, C.: The Cauchy problem for a generalized Camassa–Holm equation. Adv. Differ. Equ. 19, 161–200 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Himonas, A.A., Holliman, C., Kenig, C.: Construction of 2-peakon solutions and ill-posedness for the Novikov equation. SIAM J. Math. Anal. 50, 2968–3006 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Himonas, A.A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yin, Z.Y.: On the Cauchy problem for an integrable equation with peakon solutions. Ill. J. Math. 47, 649–666 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Zhou, Y.: On solutions to the Holm–Staley b-family of equations. Nonlinearity 23, 369–381 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ming, S., Lai, S.Y., Su, Y.Q.: Well-posedness and behaviors of solutions to an integrable evolution equation. Bound. Value Probl. 2020, 165 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kato, T.: Quasi-linear equations of evolution with applications to partial differential equations. In: Spectral Theory and Differential Equations. Lecture Notes in Math, vol. 448, pp. 25–70. Springer, Berlin (1975)

    Chapter  Google Scholar 

  19. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references


This work is supported by the National Natural Science Foundation of China (No. 11471263).

Author information

Authors and Affiliations



All authors contributed equally in this work. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Shaoyong Lai.

Ethics declarations

Competing interests

The authors declare no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, M., Tang, J. & Lai, S. A necessary and sufficient condition of blow-up for a nonlinear equation. Bound Value Probl 2023, 28 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • 35G25
  • 35L05


  • Local strong solutions
  • Nonlinear equation
  • Blow-up
  • Sufficient and necessary conditions