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

Abstract

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:

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

(1.1)

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

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

(1.2)

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 [2–4]. 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 [13–17].

We consider the following initial value problem:

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

(1.3)

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.

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

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

(1.4)

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

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

(2.1)

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

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

(2.2)

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

(2.3)

Proof

Since¹

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

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

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

(2.4)

Proof

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

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

and

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

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

([19])

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

([19])

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

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

(3.1)

then

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

(3.2)

Proof

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

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

(3.3)

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

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

(3.4)

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

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

(3.5)

Similarly to the proof of (3.5), we have

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

(3.6)

Now Lemma 2.4 yields

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

(3.7)

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

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

(3.8)

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

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

(3.9)

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

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

(3.10)

if and only if

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

(3.11)

Proof

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

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

(3.12)

Applying the first equation in (2.1) yields

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

(3.13)

Using Lemma 2.2 and (3.13), we have

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

(3.14)

Using (3.13) gives rise to

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

(3.15)

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

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

(3.16)

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

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

(3.17)

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

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

(3.18)

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

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

(3.19)

Let

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

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

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

(3.20)

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. □