Well-posedness and behaviors of solutions to an integrable evolution equation

Abstract

This work is devoted to investigating the local well-posedness for an integrable evolution equation and behaviors of its solutions, which possess blow-up criteria and persistence property. The existence and uniqueness of analytic solutions with analytic initial values are established. The solutions are analytic for both variables, globally in space and locally in time. The effects of coefficients λ and β on the solutions are given.

Introduction

We focus on investigating the following Cauchy problem:

$$\begin{aligned} \textstyle\begin{cases} w_{t}-w_{xxt} +\beta (w_{x}-w_{xxx})+\lambda (w-w_{xx})-16ww_{x} \\ \quad = 2 w_{xx}^{2}-8w_{x}w_{xx}+2w_{x}w_{xxx}-4ww_{xxx}, \\ w(0,x)=w_{0}(x). \end{cases}\displaystyle \end{aligned}$$

(1.1)

Here, $(t,x)\in \mathbb{R} ^{+}\times \mathbb{R}$, $\lambda \in \mathbb{R}^{+}$, $\beta \in \mathbb{R}$, w is the fluid velocity, $\beta (w_{x}-w_{xxx})$ is the dispersive term, $\lambda (w-w_{xx})$ is the dissipative term, $w_{0}\in B_{p,r}^{s}(\mathbb{R})(s>\max (\frac{5}{2},2+ \frac{1}{p}))$.

Problem (1.1) is viewed as a member of the integrable model

$$\begin{aligned} \bigl(1-\partial _{x}^{2}\bigr)w_{t}=F(w, w_{x}, w_{xx}, w_{xxx}), \end{aligned}$$

which has been investigated in [24]. The famous integrable Camassa–Holm (CH) equation is

$$ \bigl(1-\partial _{x}^{2}\bigr)w_{t}+3ww_{x}=- \beta w_{x}+2w_{x}w_{xx}+ww_{xxx}, $$

(1.2)

which admits peakon solutions and wave breaking mechanisms. By replacing w with $w + k$ in Eq. (1.2), Zhou and Chen [33] establish that a solution w to Eq. (1.2) may be regarded as a perturbation around the coefficient β. The wave breaking phenomena and infinite propagation speed of solutions are investigated. The behaviors of solutions to the CH equation with dissipative term and dispersion term are studied in [25]. The local well-posedness for the Cauchy problem of the CH type equations [6, 15, 20, 26, 28–31], asymptotic stability [17, 22], solitons solutions [14], and regularity of conservative solutions [18] are considered. The readers may refer to [8–10, 18, 20–22] for the related results.

Other two famous integrable models are the Degasperis–Procesi (DP) equation

$$\begin{aligned} w_{t}-w_{xxt}+4ww_{x}=3w_{x}w_{xx}+ww_{xxx} \end{aligned}$$

and the Novikov equation

$$\begin{aligned} w_{t}-w_{xxt}+4w^{2}w_{x}=3ww_{x}w_{xx}+w^{2}w_{xxx}. \end{aligned}$$

(1.3)

Molinet [23] considers the peakon solutions of the DP equation. The Novikov equation has N-peakon solutions. It is worth noticing that the first explicit 2-peakon solutions of the Novikov equation are investigated in [13]. Cai et al. [2] study the Lipschitz metric of Eq. (1.3) which possesses cubic nonlinearity. Himonas et al. [11] illustrate the construction of 2-peakon solutions and ill-posedness for the Novikov equation. The blow-up criteria of solutions to a Novikov type equation are presented in [7, 32]. The formation of singularities for solutions to problem (1.1) when $\lambda =\beta =0$ is established (see [27]). The scholars focus much attention on the CH equation and similar equations with weakly dissipative term. It is shown in [16] that some models (i.e., CH equation, DP equation, Novikov equation, and Hunter–Saxton equation) which contain weakly dissipative term can be reduced to their non-dissipative versions by applying an exponentially time-dependent scaling $u(t,x)\rightarrow e^{-\lambda t}u(\frac{1-e^{-\lambda t}}{\lambda },x)$.

To our knowledge, the influence of coefficients and properties of solutions to problem (1.1) have not been considered yet. Our study mainly focuses on investigating the influence of dissipative coefficient λ and dispersive coefficient β on the solutions to problem (1.1). We establish the blow-up criteria and blow-up rate of solutions, which are related to $n=(1-\partial _{x}^{2})w$ and dissipative coefficient λ. Moreover, the persistence properties and analytic properties of solutions are analyzed.

We define

$$ E_{p,r}^{s}(T)= \textstyle\begin{cases} C([0,T];B_{p,r}^{s}(\mathbb{R}))\cap C^{1}([0,T];B_{p,r}^{s-1}( \mathbb{R})), & 1\leq r< \infty , \\ L^{\infty } ([0,T];B_{p,\infty }^{s}(\mathbb{R}))\cap \text{Lip}([0,T];B_{p, \infty }^{s-1}(\mathbb{R})),& r=\infty , \end{cases} $$

where $T>0$, $s\in \mathbb{R}$, $p\in [1,\infty ]$, $r\in [1,\infty ]$. Problem (1.1) is written as

$$ \textstyle\begin{cases} w_{t}-4ww_{x}=-w_{x}^{2}+P_{1}(D)[2w_{x}^{2}+6w^{2}] +P_{2}(D)[w_{x}^{2} ]- \lambda w- \beta w_{x}, \\ w(0,x)=w_{0}(x), \end{cases} $$

(1.4)

where $P_{1}(D)=\partial _{x}(1-\partial _{x}^{2})^{-1}$, $P_{2}(D)=(1- \partial _{x}^{2})^{-1}$.

Let $n_{0} =(1-\partial _{x}^{2})w_{0}$ and $n =(1-\partial _{x}^{2})w$. Then problem (1.1) is reformulated as

$$ \textstyle\begin{cases} n_{t}+(2w_{x}-4w+\beta )n_{x}=2n^{2}+(8w_{x}-4w)n +2(w+w_{x})^{2}- \lambda n, \\ n(0,x)=n_{0}(x) . \end{cases} $$

(1.5)

We are in the position to summarize the main results.

Theorem 1.1

Let $1\leq p$, $r\leq \infty$, $w_{0} \in B_{p,r}^{s}(\mathbb{R}) (s>\max (\frac{5}{2}, 2+ \frac{1}{p}))$. Then a solution $w \in E_{p,r}^{s}(T)$ to problem (1.1) is unique for certain $T>0$.

Theorem 1.2

Let $1\leq p$, $r\leq \infty$, $w_{0} \in B_{p,r}^{s}(\mathbb{R}) ( \max (\frac{5}{2}, 2+ \frac{1}{p})< s<3)$, $t\in [0,T]$. Then a solution w to problem (1.1) blows up in finite time if and only if

$$\begin{aligned} \int _{0}^{t} \bigl( \bigl\Vert n(\tau ) \bigr\Vert _{L^{\infty }} -\lambda \bigr) \,d \tau =+\infty . \end{aligned}$$

Theorem 1.3

Let $1\leq p$, $r\leq \infty$, $w_{0} \in H^{s}(\mathbb{R}) (s>\frac{5}{2})$, $t\in [0,T]$. Then a solution w to problem (1.1) blows up in finite time if and only if

$$ \int _{0}^{t} \bigl( \bigl\Vert n (\tau ) \bigr\Vert _{L^{\infty }}-\lambda \bigr)\,d \tau =+\infty .$$

(1.6)

Theorem 1.4

Let $1\leq p$, $r\leq \infty$, $w_{0} \in H^{s}(\mathbb{R}) (s> \frac{5}{2})$, $n_{0} =w_{0}-w_{0,xx}$. Assume that $n_{0}(x)$ satisfies $n_{0}(x_{0})>\frac{\lambda }{2}+\sqrt{K}$, where the point $x_{0}$ is defined by $n_{0}(x_{0})=\sup_{x\in \mathbb{R}}n_{0 }(x)$, $K=\frac{\lambda ^{2}}{4}+18 \| w_{0}\| _{H^{1}}^{2}$. Let $t\in [0,T]$. Then a solution w to problem (1.1) blows up in finite time if and only if

$$ \lim_{t\rightarrow T^{-}}\biggl[\sup_{x\in \mathbb{R}} \biggl(n(t,x) - \frac{\lambda }{2}\biggr) \biggr]= +\infty .$$

(1.7)

Theorem 1.5

Let $1\leq p$, $r\leq \infty$, $w_{0} \in H^{s}(\mathbb{R}) (s> \frac{5}{2})$, $n_{0} =w_{0} -w_{0,xx}$, $t\in [0,T]$. Suppose that $[n_{0} +2w_{0,x}-w_{0}](x_{0})>\frac{\lambda }{4}+\frac{1}{2}\sqrt{K_{1}}$, where the point $x_{0}$ is defined by

$$ [n_{0} +2w_{0,x}-w_{0}](x_{0}) =\sup_{x\in \mathbb{R}}[n_{0} +2w_{0,x}-w_{0}] (x), $$

$K_{1}=2( C_{4} \| w_{0}\| _{H^{1}}^{2}+C_{5}\| w_{0} \| _{H^{1}}+C_{6})$ and $C_{4}$, $C_{5}$, $C_{6}$ are certain positive constants. Let w be a solution to problem (1.1). Then it holds that

$$\begin{aligned} \lim_{t\rightarrow T^{-}}\biggl[\sup_{x\in \mathbb{R}} \biggl(n(t,x) - \frac{\lambda }{4}\biggr) (T-t)\biggr]= \frac{1}{2}. \end{aligned}$$

Theorem 1.6

Assume $w_{0} \in H^{s}(\mathbb{R}) (s>\frac{5}{2})$, $t\in [0,T]$ and $\theta \in (0,1)$. Let $w_{0}$ satisfy

$$\begin{aligned} \bigl\vert w_{0}(x) \bigr\vert , \bigl\vert \partial _{x}w_{0}(x) \bigr\vert , \bigl\vert \partial ^{2}_{x }w_{0}(x) \bigr\vert \thicksim O \bigl(e^{-\theta x}\bigr)\quad \textit{as } x\rightarrow \infty . \end{aligned}$$

Then a solution w to problem (1.1) satisfies

$$\begin{aligned} \bigl\vert w(t,x) \bigr\vert , \bigl\vert \partial _{x}w(t,x) \bigr\vert , \bigl\vert \partial _{x}^{2}w(t,x) \bigr\vert \thicksim O\bigl(e^{-\theta x}\bigr)\quad \textit{as } x\rightarrow \infty \end{aligned}$$

uniformly on $[0,T]$.

Theorem 1.7

Let $w_{0}$ be analytic on $\mathbb{R}$ and $t\in \mathbb{R}$ in problem (1.1). Then problem (1.1) admits a unique analytic solution w on $(-\delta ,\delta ) \times \mathbb{R}$ for certain constant $\delta \in (0, 1]$.

Remark 1.1

We deduce the local well-posedness for problem (1.1) in $B_{p,r}^{s}(\mathbb{R}) (s>\max (\frac{5}{2}, 1+\frac{2}{p}))$. For presence of term $w_{x}^{2}$ in (1.4), the regularity index of solutions is $s>\max (\frac{5}{2}, 1+\frac{2}{p})$, which is different from the regularity index $s>\max (\frac{3}{2},1+\frac{1}{p})$ of solutions to the CH equation, DP equation, and Novikov equation.

Remark 1.2

We derive blow-up criterion of solutions in the Besov space in Theorem 1.2. This result is new. From Theorems 1.2, 1.3, and 1.4, we conclude that dissipative coefficient λ is related to blow-up mechanisms of solutions. From Theorem 1.4, we recognize that the blow-up phenomenon of solution w occurs if n is unbounded. From Theorem 1.5, we establish that dissipative coefficient λ is related to the precise blow-up rate of solution w. From Theorem 1.6, we observe that if initial value $w_{0}$ with its derivatives exponentially decays at infinity, then the solution w with its derivatives also exponentially decays at infinity. The existence and uniqueness of analytic solution w with analytic initial value are illustrated in Theorem 1.7. The solution w is analytic in both variables, globally in space and locally in time.

Remark 1.3

We extend parts of results in [27]. In the case $\lambda =\beta =0$ in problem (1.1), the local well-posedness for the Cauchy problem and formation of singularities of solutions are investigated in [27]. However, we mainly focus on the influence of the dispersive term and dissipative term in problem (1.1). Theorems 1.1, 1.4, and 1.5 contain the results in [27] as special cases when $\lambda =\beta =0$. In addition, for problem (1.1), we also establish blow-up criteria of solutions in the Besov space and persistence property of solutions. The existence and uniqueness of analytic solutions with analytic initial values are also studied (see detailed illustration in Remarks 1.1–1.2).

Proof of Theorem 1.1

Several lemmas

We review several basic facts in the Besov space. One may check [1] for more details.

Lemma 2.1

([1])

There exists a couple of smooth functions $(\chi (\xi ),\varphi (\xi ))$ valued in [0, 1] such that χ is supported in the ball $B= \{\xi \in \mathbb{R} | |\xi |\leq \frac{4}{3}\}$, φ is supported in the ring $C= \{\xi \in \mathbb{R} | \frac{3}{4}\leq |\xi |\leq \frac{8}{3}\}$. Moreover, it satisfies that

$$ \chi (\xi )+\sum_{q\in \mathbb{N}}\varphi \bigl(2^{-q}\xi \bigr)=1,\quad \forall \xi \in \mathbb{R} $$

and

$$\begin{aligned}& \operatorname{supp}\varphi \bigl(2^{-q}\cdot \bigr)\cap \operatorname{supp}\varphi \bigl(2^{-q'} \cdot \bigr)=\emptyset ,\quad \textit{if } \bigl\vert q-q' \bigr\vert \geq 2, \\& \operatorname{supp}\chi (\cdot )\cap \operatorname{supp}\varphi \bigl(2^{-q}\cdot \bigr)= \emptyset ,\quad \textit{if }q\geq1. \end{aligned}$$

Then, for all $u\in S'(\mathbb{R})$, the non-homogeneous dyadic blocks are defined as follows. Let

$$\begin{aligned}& \Delta _{q} u=0,\quad \textit{if } q\leq -2, \\& \Delta _{-1}u= \int _{\mathbb{R}}\chi (\xi )\widehat{u}(\xi )e^{ ix \xi }\,d\xi ,\quad \textit{if } q=-1, \\& \Delta _{q} u= \int _{\mathbb{R}}\varphi \bigl(2^{-q}\xi \bigr) \widehat{u}(\xi )e^{ ix\xi }\,d\xi ,\quad \textit{if } q\geq 0. \end{aligned}$$

Then $u=\sum_{q= -1}^{\infty }\Delta _{q} u$ is called the non-homogeneous Littlewood–Paley decomposition of u. Assume $s\in \mathbb{R}$, $1\leq p$, $r\leq \infty$. The non-homogeneous Besov space is defined by $B_{p,r}^{s}=\{f \in S'(\mathbb{R} )\mid \| f\| _{B_{p,r}^{s}}<{ \infty }\}$, where

$$ \Vert f \Vert _{B_{p,r}^{s}}= \textstyle\begin{cases} ( \sum_{j=-1}^{\infty }2^{jrs} \Vert \Delta _{j}f \Vert _{L^{p}}^{r})^{\frac{1}{r}},& r< \infty , \\ \sup_{j\geq -1}2^{js} \Vert \Delta _{j}f \Vert _{L^{p}},& r= \infty . \end{cases} $$

In addition, $S_{j}f= \sum_{q=-1}^{j-1}\Delta _{q}f$.

Lemma 2.2

([1, 5, 27])

Assume $s\in \mathbb{R}$, $1\leq p$, $r, p_{j}, r_{j}\leq \infty$, $j=1, 2$. Then

1) Embedding properties: $B_{p_{1},r_{1}}^{s}\hookrightarrow B_{p_{2},r_{2}}^{s-( \frac{1}{p_{1}}-\frac{1}{p_{2}})}$ for $p_{1}\leq p_{2}$, $r_{1}\leq r_{2}$. $B_{p ,r_{2}}^{s _{2}}\hookrightarrow B_{p ,r_{1}}^{s_{1}}$ is locally compact if $s_{1}\leq s_{2}$.

2) Algebraic properties: For all $s> 0$, $B_{p ,r }^{s }\cap L^{\infty }$ is an algebra. $B_{p ,r }^{s }$ is an algebra $\Leftrightarrow B_{p ,r }^{s }\hookrightarrow L^{\infty }$$\Leftrightarrow s>\frac{1}{p}$ or $s=\frac{1}{p}$, $r=1$.

3) Morse type estimates:

(i) Let $s>0$ and $f, g\in B_{p,r}^{s}\cap L^{\infty }$. Then there exists a positive constant C such that

$$\begin{aligned} \Vert fg \Vert _{B^{s}_{p,r}}\leq C\bigl( \Vert f \Vert _{B^{s}_{p,r}} \Vert g \Vert _{L^{\infty }}+ \Vert f \Vert _{L^{\infty }} \Vert g \Vert _{B^{s}_{p,r}}\bigr). \end{aligned}$$

(ii) For $s_{1} \leq \frac{1}{p}$, $s_{2} > \frac{1}{p}$ ($s_{2} \geq \frac{1}{p}$ if $r = 1$) and $s_{1} + s_{2} > 0$, then

$$\begin{aligned} \Vert fg \Vert _{B^{s_{1}}_{p,r}}\leq C \Vert f \Vert _{B^{s_{1}}_{p,r}} \Vert g \Vert _{B^{s_{2}}_{p,r}} . \end{aligned}$$

4) Fatou’s lemma: If a sequence $(f_{n})_{n\in \mathbb{N}}$ is bounded in $B_{p,r}^{s}$ and $f_{n}\rightarrow f$ in $S'(\mathbb{R})$, then it holds that $f\in B_{p,r}^{s}$ and

$$ \Vert f \Vert _{B_{p,r}^{s}}\leq \lim_{n\rightarrow \infty } \inf \Vert f_{n} \Vert _{B_{p,r}^{s}}. $$

5) Multiplier properties: Let $m \in \mathbb{R}$. Assume that f is an $S^{m}$-multiplier (i.e., $f:\mathbb{R} \rightarrow \mathbb{R}$ is smooth and it satisfies that, for all $\alpha \in \mathbb{N}$, there exists a positive constant $C_{\alpha }$ such that $|\partial ^{\alpha } f(\xi )|\leq C_{\alpha }(1+|\xi |)^{m-|\alpha |}$ for all $\xi \in \mathbb{R}$). Then the operator $f (D)$ is continuous from $B^{s}_{p,r}$ to $B^{s-m}_{p,r}$.

6) Density: $C_{c}^{\infty }$ is dense in $B_{p,r}^{s}\Leftrightarrow 1\leq p$, $r<\infty$.

We present two lemmas which are related to the transport equation

$$\begin{aligned} \textstyle\begin{cases} f_{t}+ d\partial _{x} f=F, \\ f|_{t=0}=f_{0}, \end{cases}\displaystyle \end{aligned}$$

(2.1)

where $d:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ represents a given time-dependent scalar function, $f_{0}:\mathbb{R} \rightarrow \mathbb{R}$ and $F:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ are the known data.

Lemma 2.3

([1])

Assume $1\leq p\leq p_{1}\leq \infty$, $1\leq r\leq \infty$, $p'= \frac{p}{p-1}$. Suppose that $s>- \min (\frac{1}{p_{1}},\frac{1}{p'})$ or $s>-1- \min (\frac{1}{p_{1}},\frac{1}{p'})$ when $\partial _{x}d=0$. Then there exists a constant $C_{1}$ depending only on p, $p_{1}$, r, s such that the following estimate holds:

$$\begin{aligned}& \Vert f \Vert _{L^{\infty }_{t}([0,t];B_{p,r}^{s})} \\& \quad \leq e^{C_{1}\int _{0}^{t} Z(\tau ) \,d \tau }\biggl[ \Vert f_{0} \Vert _{B_{p,r}^{s}}+ \int _{0}^{t} e^{-C_{1}\int _{0}^{\tau }Z(\xi ) \,d\xi } \bigl\Vert F(\tau ) \bigr\Vert _{B_{p,r}^{s}} \,d \tau \biggr], \end{aligned}$$

(2.2)

where

$$ Z(t)= \textstyle\begin{cases} \Vert \partial _{x} d(t) \Vert _{B_{p_{1},\infty }^{ \frac{1}{p_{1}}}\cap L^{\infty }},& s< 1+\frac{1}{p_{1}}, \\ \Vert \partial _{x} d(t) \Vert _{B_{p_{1},r}^{s-1}},& s>1+ \frac{1}{p_{1}} \textit{ or } s=1+\frac{1}{p_{1}},r=1. \end{cases} $$

If $f=d$, then for all $s>0$ ($s>-1$ if $\partial _{x}d=0$), (2.2) holds with $Z(t)=\| \partial _{x} d(t)\| _{L^{\infty }}$.

We present an existence result for the transport equation with initial value in the Besov space.

Lemma 2.4

([1])

Let p, $p_{1}$, r, s be in the statement of Lemma 2.3and $f_{0}\in B_{p,r}^{s}$. $F\in L^{1}([0,T];B_{p,r}^{s})$, $d \in L^{\rho }([0,T];B_{\infty , \infty }^{-M})$ is a time-dependent vector field for some $\rho >1$, $M>0$ such that if $s<1+\frac{1}{p_{1}}$, then $\partial _{x} d\in L^{1}([0,T];B_{p_{1},\infty }^{\frac{1}{p_{1}}} \cap L^{\infty })$; if $s>1+\frac{1}{p_{1}}$ or $s=1+\frac{1}{p_{1}}$, $r=1$, then $\partial _{x} d\in L^{1}([0,T];B_{p_{1},r}^{s-1})$. Therefore, problem (2.1) has a unique solution $f\in L^{\infty }([0,T];B_{p,r}^{s})\cap (\cap _{s'< s}C([0,T];B_{p,1}^{s'}))$ and (2.2) holds true. If $r<{\infty }$, it holds that $f\in C([0,T];B_{p,r}^{s})$.

Lemma 2.5

([19])

Let $1\leq p\leq \infty$, $1\leq r\leq \infty$, $s >\max (\frac{1}{2},\frac{1}{p })$. $f_{0}\in B_{p,r}^{s-1}$, $F\in L^{1} ([0,t];B_{p,r}^{s-1})$, $d\in L^{1} ([0,t];B_{p,r}^{s+1})$. Then a solution f to problem (2.1) satisfies $f\in L^{\infty }([0,T];B_{p,r}^{s-1})$ and

$$\begin{aligned}& \Vert f \Vert _{L^{\infty }_{t}([0,t];B_{p,r}^{s-1})} \\& \quad \leq e^{C_{1}\int _{0}^{t} Z(\tau ) \,d \tau }\biggl[ \Vert f_{0} \Vert _{B_{p,r}^{s-1}}+ \int _{0}^{t} e^{-C_{1}\int _{0}^{\tau }Z( \xi ) \,d\xi } \bigl\Vert F(\tau ) \bigr\Vert _{B_{p,r}^{s-1}} \,d \tau \biggr], \end{aligned}$$

where $Z(t)= \int _{0}^{t}\| d(\tau )\| _{B_{p ,r}^{s+1}} \,d \tau$, the constant $C_{1}$ depends only on s, p, and r.

Proof of Theorem 1.1

We show the framework of proof with $n_{0} \in B_{p,r}^{s} (s>\max ( \frac{1}{p},\frac{1}{2}))$.

Step 1: Set $n^{0} =0$. The smooth functions $(n^{i}) _{i\in \mathbb{N}}\in C(\mathbb{R}^{+};B_{p,r}^{\infty })$ solve the problem

$$ \textstyle\begin{cases} (\partial _{t}+(2 w_{x} ^{i}-4w^{i}+\beta )\partial _{x}) n^{i+1}=G(t,x), \\ n^{i+1}(0,x)=n^{i+1}_{0}(x)=S_{i+1}n_{0}, \end{cases} $$

(2.3)

where

$$\begin{aligned} G(t,x) =2\bigl(n^{i}\bigr)^{2}+\bigl(8w_{x}^{i}-4w^{i} \bigr)n^{i}+2\bigl(w^{i}+w_{x}^{i} \bigr)^{2}- \lambda n^{i}. \end{aligned}$$

(2.4)

Let $S_{i+1}n_{0} \in B_{p,r}^{\infty }$. In view of Lemma 2.4, we establish that $n^{i+1}\in C(\mathbb{R}^{+};B_{p,r}^{\infty })$ to problem (2.3) is global with $i\in \mathbb{N}$.

Step 2: If $s>\max \{1+\frac{1}{p},1+\frac{1}{2}\}$ or $s=\max \{1+\frac{1}{p},1+\frac{1}{2}\}$, $r=1$, we have

$$\begin{aligned} Z(t) =& \int _{0}^{t} \biggl\Vert \partial _{x}\biggl[ 2\biggl(w^{i}_{x}-2w^{i}+ \frac{1}{2}\beta \biggr) \biggr](\tau ) \biggr\Vert _{B_{p,r}^{s-1}} \,d \tau \\ =& \int _{0}^{t} \bigl\Vert \partial _{x}\bigl[ 2\bigl(w^{i}_{x}-2w^{i} \bigr) \bigr](\tau ) \bigr\Vert _{B_{p,r}^{s-1}} \,d \tau \\ \leq &C_{0} \int _{0}^{t} \bigl\Vert \bigl(w^{i}_{x}-2w^{i}\bigr) ( \tau ) \bigr\Vert _{B_{p,r}^{s }} \,d \tau \\ \leq &C_{0} \int _{0}^{t}\bigl(1+\lambda + \bigl\Vert n^{i} (\tau ) \bigr\Vert _{B_{p,r}^{s }} \bigr) \,d \tau . \end{aligned}$$

Using Lemma 2.3, we arrive at

$$\begin{aligned} \bigl\Vert n^{i+1}(t) \bigr\Vert _{B_{p,r}^{s}} \leq& e^{C_{1}\int _{0}^{t} \Vert \partial _{x}2(w^{i}_{x}-2w^{i}+\frac{1}{2}\beta ) (\tau ) \Vert _{B_{p,r}^{s-1}} \,d \tau }\times \biggl[ \Vert n_{0} \Vert _{B_{p,r}^{s}} \\ &{} + \int _{0}^{t} e^{-C_{1}\int _{0}^{\tau } \Vert \partial _{x}2(w^{i}_{x}-2w^{i}+ \frac{1}{2}\beta ) (\tau ) \Vert _{B_{p,r}^{s-1}} \,d\xi } \bigl\Vert G( \tau ,\cdot ) \bigr\Vert _{B_{p,r}^{s}} \,d \tau \biggr]. \end{aligned}$$

(2.5)

Let $a \lesssim b$ mean $a \leq Cb$ for a certain constant $C>0$. Bearing in mind the embedding property $B_{p,r}^{s} \hookrightarrow L^{\infty }(s>\max ( \frac{1}{p}, \frac{1}{2}))$, the algebra property in the Besov space and the Morse type estimate (i) in Lemma 2.2 (see [5] for more details), we acquire

$$\begin{aligned}& \bigl\Vert 2\bigl(n^{i}\bigr)^{2} \bigr\Vert _{B_{p,r}^{s}} \lesssim \bigl\Vert n^{i} \bigr\Vert _{L^{\infty }} \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s}} \lesssim \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s}}^{2}, \\& \bigl\Vert \bigl(8w_{x}^{i}-4w^{i} \bigr)n^{i} \bigr\Vert _{B_{p,r}^{s }} \\& \quad \lesssim \bigl\Vert 8w_{x}^{i}-4w^{i} \bigr\Vert _{B_{p,r}^{s }} \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s }}+ \bigl\Vert 8w_{x}^{i}-4w^{i} \bigr\Vert _{B_{p,r}^{s }} \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s }} \\& \quad \lesssim \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s}}^{2}, \\& \bigl\Vert 2\bigl(w^{i}+w_{x}^{i} \bigr)^{2} \bigr\Vert _{B_{p,r}^{s }}\lesssim \bigl\Vert w^{i}+w_{x}^{i} \bigr\Vert _{B_{p,r}^{s }}^{2}\lesssim \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s}}^{2}, \\& \bigl\Vert \lambda n^{i} \bigr\Vert _{B_{p,r}^{s}} \lesssim \lambda \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s}}. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \bigl\Vert G(t ) \bigr\Vert _{B_{p,r}^{s}}\lesssim \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s }}\bigl( 1+\lambda + \bigl\Vert n^{i} (t) \bigr\Vert _{B_{p,r}^{s }} \bigr). \end{aligned}$$

(2.6)

It is worth noticing that

$$\begin{aligned} \bigl\Vert \partial _{x}\bigl(w^{i}_{x}-2w^{i} \bigr) (\tau ) \bigr\Vert _{B_{p,r}^{s-1}} \lesssim 1+\lambda + \bigl\Vert n^{i} (\tau ) \bigr\Vert _{B_{p,r}^{s }}. \end{aligned}$$

(2.7)

Combining (2.5) with (2.7), we deduce

$$\begin{aligned} \bigl\Vert n^{i+1}(t) \bigr\Vert _{B_{p,r}^{s}} \lesssim &e^{C_{1}\int _{0}^{t} \Vert \partial _{x}(w^{i}_{x}-2w^{i}) (\tau ) \Vert _{B_{p,r}^{s-1}} \,d \tau } \Vert n_{0} \Vert _{B_{p,r}^{s}} \\ & {}+ \int _{0}^{t} e^{ C_{1}\int _{\tau }^{t} \Vert \partial _{x}(w^{i}_{x}-2w^{i}) (\xi ) \Vert _{B_{p,r}^{s-1}} \,d\xi } \bigl\Vert G(\tau ,\cdot ) \bigr\Vert _{B_{p,r}^{s}} \,d \tau \\ \lesssim& e^{C_{2}\int _{0}^{t} (1+\lambda + \Vert n^{i}(\tau ) \Vert _{B_{p,r}^{s}} ) \,d \tau } \Vert n_{0} \Vert _{B_{p,r}^{s}} \\ & {}+ \int _{0}^{t} e^{ C_{2}\int _{\tau }^{t} (1+\lambda + \Vert n^{i}(\xi ) \Vert _{B_{p,r}^{s}} ) \,d\xi } \bigl\Vert G( \tau ,\cdot ) \bigr\Vert _{B_{p,r}^{s}} \,d \tau . \end{aligned}$$

(2.8)

Plugging (2.6) into (2.8) leads to the inequality

$$\begin{aligned} \bigl\Vert n^{i+1}(t) \bigr\Vert _{B_{p,r}^{s}} \leq& C_{2}\cdot e^{C_{2} \int _{0}^{t} (1+\lambda + \Vert n^{i}(\tau ) \Vert _{B_{p,r}^{s}} ) \,d \tau } \biggl[ \Vert n_{0} \Vert _{B_{p,r}^{s}} \\ &{} + \int _{0}^{t}e^{-C_{2} \int _{0}^{\tau }(1+\lambda + \Vert n^{i}(\xi ) \Vert _{B_{p,r}^{s}} ) \,d\xi } \\ &{}\times \bigl(1+\lambda + \bigl\Vert n^{i}(\tau ) \bigr\Vert _{B_{p,r}^{s}} \bigr) \bigl\Vert n^{i}(\tau ) \bigr\Vert _{B_{p,r}^{s}} \,d \tau \biggr]. \end{aligned}$$

(2.9)

If $\max \{ \frac{1}{p}, \frac{1}{2}\}< s<\max \{1+\frac{1}{p},1+ \frac{1}{2}\}$, applying the embedding property $B_{p,r}^{s}\hookrightarrow L^{\infty }$, we have

$$\begin{aligned} Z(t) =& \int _{0}^{t} \biggl\Vert \partial _{x}\biggl[ 2\biggl(w^{i}_{x}-2w^{i}+ \frac{1}{2}\beta \biggr) \biggr](\tau ) \biggr\Vert _{B_{p,\infty }^{\frac{1}{p}} \cap L^{\infty }} \,d \tau \\ \lesssim& \int _{0}^{t} \bigl\Vert \bigl[ \partial _{x}\bigl(w^{i}_{x}-2w^{i} \bigr) \bigr](\tau ) \bigr\Vert _{B_{p,r}^{s }} \,d \tau \\ \lesssim& \int _{0}^{t} \bigl\Vert \bigl(w^{i}_{x}-2w^{i}\bigr) (\tau ) \bigr\Vert _{B_{p,r}^{s+1 }} \,d \tau \lesssim \int _{0}^{t} \bigl(1+\lambda + \bigl\Vert n^{i} (\tau ) \bigr\Vert _{B_{p,r}^{s }}\bigr) \,d \tau . \end{aligned}$$

Similarly, we deduce that (2.9) holds true in this case.

Therefore, one can choose certain $T>0$ to satisfy $2C_{2}^{2}(1+\lambda +\| n_{0}\| _{B_{p,r}^{s}} )T<1$ and

$$ 1+\lambda + \bigl\Vert n^{i}(t) \bigr\Vert _{B_{p,r}^{s}} \leq \frac{C_{2}( 1+\lambda + \Vert n_{0} \Vert _{B_{p,r}^{s}} )}{1-2C_{2} ^{2}( 1+\lambda + \Vert n_{0} \Vert _{B_{p,r}^{s}} )t}, $$

(2.10)

which combined with (2.9) results in

$$ 1+\lambda + \bigl\Vert n^{i+1}(t) \bigr\Vert _{B_{p,r}^{s}} \leq \frac{C_{2}( 1+\lambda + \Vert n_{0} \Vert _{B_{p,r}^{s}} )}{1-2C_{2}^{2}( 1+\lambda + \Vert n_{0} \Vert _{B_{p,r}^{s}} )t}. $$

We achieve that $(n^{i})_{i\in \mathbb{N}}$ is uniformly bounded in $E_{p,r}^{s}(T)$.

Step 3: Utilizing problem (2.3) gives rise to

$$\begin{aligned}& \bigl(\partial _{t}+\bigl(2 w_{x}^{i+j}-4w^{i+j}+ \beta \bigr)\partial _{x}\bigr) \bigl(n^{i+j+1}-n^{i+1} \bigr) \\& \quad =-\bigl[2\bigl( w^{i+j}_{x} -w^{i}_{x} \bigr)-4\bigl(w^{i+j}-w^{i}\bigr)\bigr]n^{i+1}_{x} \\& \qquad{} +2\bigl(n^{i+j}+n^{i}\bigr) \bigl(n^{i+j}-n^{i}\bigr)+\bigl(8w_{x}^{i+j}-4w^{i+j} \bigr) \bigl(n^{i+j}-n^{i}\bigr) \\& \qquad{} +\bigl(8\bigl(w_{x}^{i+j}-w_{x}^{i } \bigr)-4\bigl(w ^{i+j}-w^{i }\bigr)\bigr)n^{i } \\& \qquad{} +2\bigl(w ^{i+j}+v ^{i+j}_{x}+w^{i }+w^{i }_{x} \bigr) \bigl(w ^{i+j}-w^{i }+w^{i+j}_{x}-w ^{i }_{x}\bigr) \\& \qquad{} -\lambda \bigl(n^{i+j}-n^{i}\bigr). \end{aligned}$$

(2.11)

Thanks to Lemma 2.5, we acquire

$$\begin{aligned}& \bigl\Vert n^{i+j+1}-n^{i+1} \bigr\Vert _{B_{p,r}^{s-1}}\\& \quad \leq e^{C \int _{0}^{t} \Vert n^{i+j} \Vert _{B_{p,r}^{s}} \,d \tau }\biggl[ \bigl\Vert n_{0}^{i+j+1}-n_{0}^{i+1} \bigr\Vert _{B_{p,r}^{s-1}} \\& \qquad {} +C \int _{0}^{t} e^{-C \int _{0}^{\tau } \Vert n^{i+j} \Vert _{B_{p,r}^{s}} \,d\xi } \bigl\Vert n^{i+j}-n^{i} \bigr\Vert _{B_{p,r}^{s-1}} \\& \qquad {} \times \bigl(1+\lambda + \bigl\Vert n^{i} \bigr\Vert _{B_{p,r}^{s}}+ \bigl\Vert n^{i+j} \bigr\Vert _{B_{p,r}^{s}}+ \bigl\Vert n^{i+1} \bigr\Vert _{B_{p,r}^{s}} \bigr) \,d \tau \biggr]. \end{aligned}$$

Since

$$ n_{0}^{i+j+1}-n_{0}^{i+1}=\sum _{q=i+1}^{i+j}\Delta _{q} w_{0}, $$

we can choose a constant $C_{1}>0$ to satisfy

$$\begin{aligned} \bigl\Vert n^{i+j+1}-n^{i+1} \bigr\Vert _{L^{\infty }([0,T];B_{p,r}^{s-1})} \leq C_{1}2^{-i} . \end{aligned}$$

As a consequence, we derive that $(n^{i} )_{i\in \mathbb{N}}$ is a Cauchy sequence in $C([0,T];B_{p,r}^{s-1})$.

Step 4: Existence of solutions.

Using the Fatou property in Lemma 2.2 yields that $n \in L^{\infty }([0,T];B_{p,r}^{s})$. It is worth noticing that $(n^{i} )_{i\in \mathbb{N}}$ is a Cauchy sequence in $C([0,T];B_{p,r}^{s-1})$ which converges to a limit function $n \in C([0,T];B_{p,r}^{s-1})$. Making use of an interpolation argument yields that the convergence holds in $C([0,T];B_{p,r}^{s'})$ for all $s'< s$. Sending $i\rightarrow \infty$ in (2.3) yields that n is a solution to (2.3). Then the right-hand side of the first equation in (2.3) belongs to $L^{\infty }([0,T];B_{p,r}^{s})$. In the case $r<\infty$, taking advantage of Lemma 2.4 gives rise to $n\in C([0,T];B_{p,r}^{s'})$ for all $s'< s$.

Applying (1.5) yields that $n_{t} \in C([0,T];B_{p,r}^{s-1})$ if $r<\infty$, and $n_{t} \in L^{\infty }([0,T];B_{p,r}^{s-1})$ otherwise. Thus, $n \in E_{p,r}^{s}(T)$. Employing a sequence of viscosity approximate solutions $(n_{\varepsilon })_{\varepsilon >0}$ to problem (1.5) which converges uniformly in $C([0,T];B_{p,r}^{s})\cap C^{1}([0,T];B_{p,r}^{s-1})$, we achieve the continuity of solution $n \in E_{p,r}^{s}(T)$.

Step 5: Uniqueness and continuity with respect to initial data.

We assume that $n^{1}$ and $n^{2}$ are two given solutions to problem (1.5) with initial values $n_{0}^{1}, n_{0}^{2} \in B_{p,r}^{s}$. $n^{1},n^{2}\in L^{\infty }([0,T];B_{p,r}^{s})\cap C([0,T];B_{p,r}^{s-1})$ and $n^{12}=n^{1}-n^{2}$. Then it holds that

$$ \textstyle\begin{cases} (\partial _{t}+(2 w_{x}^{1}-4w^{1}+\beta )\partial _{x}) n^{12} =-(2 w^{12}_{x} -4 w^{12} ) n^{1}_{x}+G_{1}(t,x) , \\ n^{12}(0,x)=n_{0}^{12}=n_{0}^{1}-n_{0}^{2}, \end{cases} $$

(2.12)

where

$$\begin{aligned} G_{1}(t,x) =&2\bigl(n^{1}+n^{2}\bigr) n^{12} +\bigl(8w_{x}^{1}-4w^{1} \bigr) n^{12} + \bigl(8 w_{x}^{12} -4 w ^{12} \bigr)n^{2} \\ &{} +2\bigl(w ^{1}+w ^{1}_{x}+w ^{2 }+w ^{2 }_{x}\bigr) \bigl(w ^{12} +w ^{12}_{x} \bigr)-\lambda n^{12}. \end{aligned}$$

In view of Lemma 2.5, we deduce

$$\begin{aligned}& \begin{gathered}[b] e^{-C\int _{0}^{t} \Vert 2 w_{x}^{1}-4w^{1} \Vert _{B_{p,r}^{s+1}} \,d \tau } \bigl\Vert n^{12}(t) \bigr\Vert _{B_{p,r}^{s-1}} \\ \quad \leq \bigl\Vert n^{12}_{0} \bigr\Vert _{B_{p,r}^{s-1}} +C \int _{0}^{t} e^{-C \int _{0}^{\tau } \Vert 2 w_{x}^{1}-4w^{1} \Vert _{B_{p,r}^{s+1}} \,d\xi } \\ \qquad {}\times \bigl( \bigl\Vert -\bigl(2 w^{12}_{x} -4 w^{12} \bigr) n^{1}_{x} \bigr\Vert _{B_{p,r}^{s-1}}+ \bigl\Vert G_{1}(\tau ) \bigr\Vert _{B_{p,r}^{s-1}}\bigr) \,d \tau . \end{gathered} \end{aligned}$$

(2.13)

Taking advantage of the Morse type estimates in Lemma 2.2 and applying $s>\max ( \frac{1}{p},\frac{1}{2})$, we have

$$\begin{aligned} \bigl\Vert -\bigl(2 w^{12}_{x} -4 w^{12} \bigr) n^{1}_{x} \bigr\Vert _{B_{p,r}^{s-1}} \lesssim& \bigl\Vert -\bigl(2 w^{12}_{x} -4 w^{12} \bigr) \bigr\Vert _{B_{p,r}^{s }} \bigl\Vert n^{1}_{x} \bigr\Vert _{B_{p,r}^{s-1}} \\ \lesssim &\bigl\Vert n^{12} \bigr\Vert _{B_{p,r}^{s-1 }} \bigl\Vert n^{1} \bigr\Vert _{B_{p,r}^{s }}. \end{aligned}$$

Similarly, we acquire

$$\bigl\Vert G_{1}(t) \bigr\Vert _{B_{p,r}^{s-1}}\lesssim \bigl\Vert n^{12} \bigr\Vert _{B_{p,r}^{s-1}}\bigl( 1+\lambda + \bigl\Vert n^{1} \bigr\Vert _{B_{p,r}^{s}}+ \bigl\Vert n^{2} \bigr\Vert _{B_{p,r}^{s}}\bigr). $$

Direct computation shows that

$$\begin{aligned}& e^{-C\int _{0}^{t} \Vert n^{1} \Vert _{B_{p,r}^{s }} \,d \tau } \bigl\Vert n^{12} \bigr\Vert _{B_{p,r}^{s-1}} \\& \quad \leq \bigl\Vert n^{12}_{0} \bigr\Vert _{B_{p,r}^{s-1}} +C \int _{0}^{t} e^{-C\int _{0}^{\tau } \Vert n^{1} \Vert _{B_{p,r}^{s }} \,d\xi } \bigl\Vert n^{12} \bigr\Vert _{B_{p,r}^{s-1}} \\& \qquad {}\times \bigl( 1+\lambda + \bigl\Vert n^{1} \bigr\Vert _{B_{p,r}^{s}}+ \bigl\Vert n^{2} \bigr\Vert _{B_{p,r}^{s}} \bigr) \,d \tau . \end{aligned}$$

Making use of the Gronwall inequality yields

$$\begin{aligned} e^{-C\int _{0}^{t} \Vert n^{1} \Vert _{B_{p,r}^{s }} \,d \tau } \bigl\Vert n^{12} \bigr\Vert _{B_{p,r}^{s-1}} \leq \bigl\Vert n^{12}_{0} \bigr\Vert _{B_{p,r}^{s-1}}e^{ \int _{0}^{t} ( 1+\lambda + \Vert n^{1} \Vert _{B_{p,r}^{s}}+ \Vert n^{2} \Vert _{B_{p,r}^{s}} )\,d \tau }. \end{aligned}$$

It follows that

$$\begin{aligned} \bigl\Vert n^{12} \bigr\Vert _{B_{p,r}^{s-1}}\leq \bigl\Vert n^{12}_{0} \bigr\Vert _{B_{p,r}^{s-1}} e^{ C\int _{0}^{t} \Vert n^{1} \Vert _{B_{p,r}^{s}} \,d \tau } e^{ \int _{0}^{t} ( 1+\lambda + \Vert n^{1} \Vert _{B_{p,r}^{s}}+ \Vert n^{2} \Vert _{B_{p,r}^{s}} )\,d \tau }. \end{aligned}$$

(2.14)

From step 2 in this section, we observe that $\| n^{1}\| _{B_{p,r}^{s}}$ and $\| n^{2}\| _{B_{p,r}^{s}}$ are uniformly bounded for all $t\in (0,T]$.

Therefore, $e^{ C\int _{0}^{t} \| n^{1}\| _{B_{p,r}^{s}} \,d \tau }$ and $e^{ \int _{0}^{t} ( 1+\lambda +\| n^{1}\| _{B_{p,r}^{s}}+ \| n^{2}\| _{B_{p,r}^{s}} )\,d \tau }$ in (2.14) are bounded for all $t\in (0,T]$. In particular, if $n_{0}^{1}=n_{0}^{2}$, we have $n_{0}^{12}(x)=n_{0}^{1}-n_{0}^{2}=0$ for $x\in \mathbb{R}$. It is deduced from (2.14) that $\| n^{12}\| _{B_{p,r}^{s-1}}\leq 0$ for all $t\in (0,T]$. It follows that $n^{12}(t,x)=n^{1}-n^{2}=0$ for all $t\in (0,T]$, $x\in \mathbb{R}$.

Thus, we arrive at the desired results.

Remark 2.1

When $p= r = 2$, the Besov space $B_{p,r}^{s} (\mathbb{R})$ coincides with the Sobolev space $H^{s} (\mathbb{R})$. It is worth noticing that $(1-\partial _{x}^{2})^{-1}$ is an $S^{-2}$ multiplier. Then it holds that

$$\begin{aligned} \Vert w \Vert _{B_{p,r}^{s+2}}= \bigl\Vert \bigl(1-\partial _{x}^{2}\bigr)^{-1} \bigl(1-\partial _{x}^{2}\bigr) w \bigr\Vert _{B_{p,r}^{s+2}}\lesssim \bigl\Vert \bigl(1- \partial _{x}^{2} \bigr)^{-1} n \bigr\Vert _{B_{p,r}^{s+2}}\lesssim \Vert n \Vert _{B_{p,r}^{s }}. \end{aligned}$$

Theorem 1.1 indicates that under the assumption $w_{0} \in H^{s} (\mathbb{R})(s>\frac{5}{2})$, we establish the local well-posedness for problem (1.1) and the solution satisfies $w\in C([0,T];H^{s}(\mathbb{R}))\cap C^{1}([0,T];H^{s-1}(\mathbb{R}))$.

Remark 2.2

Let $1\leq p$, $r\leq \infty$ and $w_{0} \in B_{p,r}^{s} (\mathbb{R})(s>\max (\frac{5}{2},2+ \frac{1}{p}))$. Then a solution w to problem (1.1) satisfies the inequality

$$ \bigl\Vert w(t) \bigr\Vert _{H^{1}}\leq \Vert w_{0} \Vert _{H^{1}},\quad t\in [0,T].$$

(2.15)

Proofs of Theorems 1.2, 1.3, 1.4, and 1.5

We recall a lemma which is related to the commutator estimates.

Lemma 3.1

([1])

Assume $s>0$, $1\leq p\leq p_{1}\leq \infty$, $1\leq r\leq \infty$, $\frac{1}{p_{2}}=\frac{1}{p }-\frac{1}{p_{1}}$. f and g are scalar functions on $\mathbb{R}$. Then

$$\begin{aligned} \bigl\Vert [\Delta _{j}, f\partial _{x}]g \bigr\Vert _{B_{p,r}^{s}} \lesssim \Vert \partial _{x} f \Vert _{L^{\infty }} \Vert g \Vert _{B_{p,r}^{s}}+ \Vert \partial _{x} f \Vert _{B_{p_{1},r}^{s-1}} \Vert \partial _{x} g \Vert _{L^{p_{2}}} \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert [\Delta _{j}, f\partial _{x}]g \bigr\Vert _{B_{p,r}^{s}}\leq C \Vert \partial _{x} f \Vert _{L^{\infty }} \Vert g \Vert _{B_{p,r}^{s}} \quad \textit{with } 0< s< 1 . \end{aligned}$$

Proof of Theorem 1.2

Applying the operator $\Delta _{q}$ to problem (1.5) leads to

$$\begin{aligned} \bigl(\partial _{t}+(2w_{x}-4w+\beta )\partial _{x}\bigr)\Delta _{q} n= [2w_{x}-4w, \Delta _{q}]\partial _{x}n+\Delta _{q} G_{2}(t,x)-\lambda \Delta _{q} n , \end{aligned}$$

(3.1)

where

$$\begin{aligned} G_{2}(t,x)=2n^{2}+(8w_{x}-4w)n+2(w+w_{x})^{2}. \end{aligned}$$

Utilizing $n_{0}\in B_{p,r}^{s}(\mathbb{R}) (\max (\frac{1}{2}, \frac{1}{p})< s<1)$ and Lemma 3.1, it yields

$$\begin{aligned}& \bigl\Vert \bigl[\Delta _{q}, (2w_{x}-4w) \partial _{x}\bigr]n \bigr\Vert _{B_{p,r}^{s }} \\& \quad \lesssim \bigl\Vert \partial _{x}(2w_{x}-4w) \bigr\Vert _{L^{\infty }} \Vert n \Vert _{B_{p,r}^{s }} \\& \quad \lesssim \Vert n \Vert _{L^{\infty }} \Vert n \Vert _{B_{p,r}^{s }} \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert G_{2}(t,x) \bigr\Vert _{B_{p,r}^{s }} \lesssim& \bigl\Vert 2n^{2}+(8w_{x}-4w)n+2(w+w_{x})^{2}- \lambda n \bigr\Vert _{B_{p,r}^{s }} \\ \lesssim& \Vert n \Vert _{L^{\infty }} \Vert n \Vert _{B_{p,r}^{s }} \\ &{} + \Vert 8w_{x}-4w \Vert _{L^{\infty }} \Vert n \Vert _{B_{p,r}^{s }} + \Vert 8w_{x}-4w \Vert _{B_{p,r}^{s }} \Vert n \Vert _{L^{\infty }} \\ & {}+ \Vert w_{x}+w \Vert _{L^{\infty }} \Vert w_{x}+w \Vert _{B_{p,r}^{s }} \\ \lesssim& \Vert n \Vert _{L^{\infty }} \Vert n \Vert _{B_{p,r}^{s}}. \end{aligned}$$

Multiplying (3.1) by $(\Delta _{q} n)^{p-1}$ and integrating on $\mathbb{R}$, we acquire

$$\begin{aligned} \frac{1}{p}\,\frac{ d}{ dt} \Vert \Delta _{q}n \Vert _{L^{p}}^{p} \lesssim& \bigl\Vert \partial _{x}(2w_{x}-4w+\beta ) \bigr\Vert _{L^{ \infty }} \Vert \Delta _{q}n \Vert _{L^{p}}^{p} \\ &{} + \bigl\Vert [2w_{x}-4w,\Delta _{q}] \partial _{x}n \bigr\Vert _{L^{p}} \Vert \Delta _{q}n \Vert _{L^{p}}^{p-1} \\ &{} + \bigl\Vert \Delta _{q} G_{2}(t,x) \bigr\Vert _{L^{p}} \Vert \Delta _{q}n \Vert _{L^{p}}^{p-1}-\lambda \Vert \Delta _{q}n \Vert _{L^{p}}^{p}. \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} \frac{d}{dt} \Vert \Delta _{q}n \Vert _{L^{p}} \lesssim& \bigl\Vert \partial _{x}(2w_{x}-4w+ \beta ) \bigr\Vert _{L^{\infty }} \Vert \Delta _{q}n \Vert _{L^{p}} \\ &{} + \bigl\Vert [2w_{x}-4w,\Delta _{q}] \partial _{x}n \bigr\Vert _{L^{p}} + \bigl\Vert \Delta _{q} G_{2}(t,x) \bigr\Vert _{L^{p}}- \lambda \Vert \Delta _{q}n \Vert _{L^{p}}. \end{aligned}$$

Making use of Lemma 2.1 gives rise to

$$\begin{aligned} \bigl\Vert n(t) \bigr\Vert _{B_{p,r}^{s }} \lesssim \Vert n_{0} \Vert _{B_{p,r}^{s }} + \int _{0}^{t} \bigl( \bigl\Vert n(\tau ) \bigr\Vert _{L^{\infty }}-\lambda \bigr) \bigl\Vert n(\tau ) \bigr\Vert _{B_{p,r}^{s }} \,d \tau . \end{aligned}$$

Applying the Gronwall inequality, we conclude

$$\begin{aligned} \bigl\Vert n(t) \bigr\Vert _{B_{p,r}^{s }} \lesssim \Vert n_{0} \Vert _{B_{p,r}^{s }} e^{\int _{0}^{t} ( \Vert n(\tau ) \Vert _{L^{\infty }}-\lambda ) \,d \tau }. \end{aligned}$$

(3.2)

Suppose that $T^{\ast }<\infty$ is the maximal existence time of solutions to problem (1.5). If

$$\begin{aligned} \int _{0}^{t} \bigl( \bigl\Vert n(\tau ) \bigr\Vert _{L^{\infty }}-\lambda \bigr) \,d \tau < \infty , \end{aligned}$$

(3.3)

we acquire that $\| n(T^{\ast })\| _{B_{p,r}^{s }}$ is bounded in view of (3.2). The proof of Theorem 1.2 is completed.

Proof of Theorem 1.3

We illustrate the proof with density argument in the case $s=3$. Due to problem (1.5), we acquire the identity

$$\begin{aligned}& \frac{1}{2}\,\frac{ d}{ dt} \int _{\mathbb{R}} n^{2} \,dx = \int _{\mathbb{R}} \bigl[(n+6w_{x}-3w)n^{2}+2(w+w_{x})^{2}n-( \beta n_{x}+ \lambda n)n\bigr] \,dx. \end{aligned}$$

(3.4)

That is,

$$\begin{aligned} \frac{1}{2}\,\frac{ d}{ dt} \Vert n \Vert _{H^{1}} ^{2} =& \frac{1}{2}\,\frac{ d}{ dt} \bigl( \Vert n \Vert _{L^{2}} ^{2}+ \Vert n_{x} \Vert _{L^{2}} ^{2} \bigr) \\ =& \int _{\mathbb{R}} \bigl[(n+6w_{x}-3w)n^{2}+2(w+w_{x})^{2}n-( \beta n_{x}+ \lambda n)n\bigr] \,dx \\ & {} + \int _{\mathbb{R}} \bigl[5(2w_{x}+n-w)n_{x}^{2}+4(w-2w_{x})nn_{x} \bigr]\,dx \\ & {} - \int _{\mathbb{R}} 8\bigl[(w+w_{x})^{2}n-(w+uw_{x})n^{2} \bigr]\,dx+ \int _{\mathbb{R}} \bigl[-(\beta n_{xx}+\lambda n_{x})n_{x}\bigr] \,dx \\ \lesssim& \Vert n+6w_{x}-3w \Vert _{L^{\infty }} \Vert n \Vert _{L^{2}}^{2}+ \Vert w+w_{x} \Vert _{L^{\infty }}^{2} \Vert n \Vert _{L^{1}} \\ &{} + \Vert 2w_{x}+n-w \Vert _{L^{\infty }} \Vert n_{x} \Vert _{L^{2}}^{2}+ \Vert w-2w_{x} \Vert _{L^{\infty }} \Vert n \Vert _{L^{\infty }} \Vert n_{x} \Vert _{L^{1}} \\ &{} + \Vert w+w_{x} \Vert _{L^{\infty }}^{2} \Vert n \Vert _{L^{1}} + \Vert w+w_{x} \Vert _{L^{\infty }} \Vert n \Vert _{L^{2}}^{2}-\lambda \Vert n \Vert _{H^{1}}^{2} \\ \lesssim &\bigl( \Vert n \Vert _{L^{\infty }}-\lambda \bigr) \Vert n \Vert _{H^{1}}^{2}. \end{aligned}$$

(3.5)

Eventually, we deduce

$$\begin{aligned} \bigl\Vert n(t) \bigr\Vert _{H^{1}}\lesssim \Vert n_{0} \Vert _{H^{1}} e^{ \int _{0}^{t} ( \Vert n (\tau ) \Vert _{L^{\infty }}- \lambda )\,d \tau }, \end{aligned}$$

(3.6)

which yields a contradiction.

Proof of Theorem 1.4

Lemma 3.2

([4])

Let $T>0$, $w\in C^{1}([0,T);H^{3}(\mathbb{R}))$ and $n=(1-\partial _{x}^{2})w$. Then, for all $t\in [0,T)$, there exists one point $\xi (t)\in \mathbb{R}$ such that

$$\begin{aligned} n_{1}(t)=\sup_{x\in \mathbb{R}}n(t,x)=n\bigl(t,\xi (t) \bigr) \end{aligned}$$

(3.7)

and

$$\begin{aligned} \frac{ d}{ dt} n_{1}(t)=n_{1,t}\bigl(t,\xi (t) \bigr), \end{aligned}$$

where $n_{1}(t)$ is absolutely continuous on $(0,T)$.

Consider the problem

$$ \textstyle\begin{cases} \frac{d}{dt}p(t,x)=(2w_{x}-4w )(t,p(t,x))+\beta , \\ p(0,x)=x. \end{cases} $$

(3.8)

Lemma 3.3

([3])

Let $w\in C([0,T];H^{s}(\mathbb{R}))\cap C^{1}([0,T];H^{s-1}(\mathbb{R}))(s \geq 3)$, $n=w-w_{xx}$. Then problem (3.8) admits a unique solution $p(t,x)\in C^{1}([0,T]\times \mathbb{R}, \mathbb{R})$. Moreover, the map $p(t,\cdot )$ is an increasing diffeomorphism of $\mathbb{R}$ for all $t \in [0, T )$ and $p(t,x)$ satisfies the equality

$$\begin{aligned} p_{x}(t,x)=e^{\int _{0}^{t}(2w-2n-4w_{x})(\tau ,p(\tau ,x)) \,d \tau }. \end{aligned}$$

(3.9)

Lemma 3.4

Let $w_{0}\in H^{s}(\mathbb{R})(s\geq 3)$, $n_{0}=w_{0}-w_{0,xx}$, $(t,x)\in [0,T]\times \mathbb{R}$. Then

$$\begin{aligned} n\bigl(t,p(t,x)\bigr)p_{x}^{2}(t,x)\geq n_{0}(x)e^{\int _{0}^{t}(-2n(\tau ,p( \tau ,x))-\lambda ) \,d \tau } . \end{aligned}$$

(3.10)

Proof of Lemma 3.4

Utilizing (3.8) and Lemma 3.3 gives rise to

$$\begin{aligned} \frac{d}{dt}\bigl[n\bigl(t,p(t,x)\bigr)p_{x}^{2}(t,x) \bigr] =&(n_{t}+n_{x}p_{t})p_{x}^{2}+2n p_{x} p_{xt} \\ =&p_{x}^{2}\bigl[2(w+w_{x})^{2}-2n^{2} \bigr] -\lambda np_{x}^{2} \\ \geq& (-2n-\lambda ) n p_{x}^{2}. \end{aligned}$$

Making use of the Gronwall inequality, we complete the proof of Lemma 3.4. □

Proof of Theorem 1.4

We present the proof by using Lemmas 3.2–3.4 with density argument in the case $s=3$. Taking advantage of the assumption $n_{0}(x)> 0$ and Lemma 3.4 yields $n(t,x) > 0$. In view of $w(t,x)=g*n$ and $g(x)=\frac{1}{2}e^{-|x|}$, it satisfies

$$\begin{aligned} w(t,x)=\frac{1}{2} \int _{\mathbb{R}}e^{-|x-\xi |} n(t,\xi ) \,d\xi \geq 0. \end{aligned}$$

It follows that

$$\begin{aligned} w(t,x)=\frac{1}{2}e^{-x} \int _{-\infty }^{x} e^{\xi }n(t,\xi ) \,d\xi + \frac{1}{2}e^{x} \int _{x}^{+\infty } e^{-\xi }n(t,\xi ) \,d\xi \end{aligned}$$

(3.11)

and

$$\begin{aligned} w_{x}(t,x)=-\frac{1}{2}e^{-x} \int _{-\infty }^{x} e^{\xi }n(t,\xi ) \,d\xi +\frac{1}{2}e^{x} \int _{x}^{+\infty } e^{-\xi }n(t,\xi ) \,d\xi . \end{aligned}$$

(3.12)

Thus we conclude $|w_{x} |\leq w$ and

$$ n_{t}+(2w_{x}-4w+\beta )n_{x} \geq n^{2} -\lambda n-18 \Vert w_{0} \Vert _{H^{1}}^{2}, $$

(3.13)

where we have used Remark 2.2 and

$$(4w_{x}-2w)^{2}\leq 36w^{2} \leq 36 \biggl( \frac{1}{\sqrt{2}} \Vert w \Vert _{H^{1}}\biggr)^{2} \leq 18 \Vert w_{0} \Vert _{H^{1}} ^{2}. $$

Set $n_{1} (t)=\sup_{x\in \mathbb{R}}[n(t,x)]$. Applying Lemma 3.2, we deduce that there exists $\xi (t), t\in [0,T)$ such that

$$\begin{aligned} n_{1}(t)=\sup_{x\in \mathbb{R}}n(t,x)=n \bigl(t,\xi (t) \bigr) . \end{aligned}$$

Thus, we come to $n_{x}(t,\xi (t))=0$.

We recall that $p(t,\cdot ):\mathbb{R}\rightarrow \mathbb{R}$ is a diffeomorphism for all $t\in [0,T)$. There exists $x_{1}(t)\in \mathbb{R}$ such that $p(t,x_{1}(t))=\xi (t)$. From (3.13), we acquire

$$\begin{aligned} \frac{d}{dt}n_{1} (t)\geq n_{1} ^{2} -\lambda n_{1} -18 \Vert w_{0} \Vert _{H^{1}}^{2}. \end{aligned}$$

(3.14)

Setting

$$\begin{aligned} n_{2}(t)=-\biggl[n_{1}(t)-\frac{\lambda }{2}\biggr] \quad \text{and}\quad K= \frac{\lambda ^{2}}{4}+18 \Vert w_{0} \Vert _{H^{1}}^{2}, \end{aligned}$$

(3.15)

we have

$$\begin{aligned} \frac{d}{dt}\bigl[ n_{2}(t) \bigr] \leq -\bigl[ n_{2}(t)\bigr] ^{2} +K. \end{aligned}$$

(3.16)

Then $n_{2}(t)$ is strictly decreasing on $[0,T)$.

Recalling the condition $n_{0}(x_{0})>\frac{\lambda }{2}+\sqrt{K}$ with $x_{0}$ defined by $n(x_{0})=\sup_{x\in \mathbb{R}}n_{0 }(x)$ in Theorem 1.4 and letting $\xi (0)=x_{0}$, we deduce $n _{2}(0)=-(n_{1}(0)-\frac{\lambda }{2})=-(n_{0}(\xi (0))- \frac{\lambda }{2}) =-(n_{0}(x_{0})-\frac{\lambda }{2})<-\sqrt{K}$. We choose $\delta \in (0,1)$ to satisfy $-\sqrt{\delta } n_{2}(0) =\sqrt{ K}$.

Utilizing (3.16), we observe

$$\begin{aligned} \frac{ d }{ dt} \biggl(\frac{1 }{n_{2}(t)} \biggr)=- \frac{1}{n_{2}^{2}(t)}\frac{ d n_{2}(t)}{ dt}\geq 1-\delta . \end{aligned}$$

(3.17)

That is,

$$\begin{aligned} -\frac{1}{n_{2}(t)}+\frac{1}{n_{2}(0)}\leq -(1-\delta )t. \end{aligned}$$

(3.18)

Bearing in mind $n_{2}(t)<0$, $t\in [0,T]$, we come to the estimate $T\leq \frac{-1}{(1-\delta )n_{2}(0)}<\infty$, where $n_{2}(0)=-(n_{0}(x_{0})-\frac{\lambda }{2}) <0$. It turns out that

$$\begin{aligned}& -\biggl[n \bigl(t,\xi (t )\bigr)-\frac{\lambda }{2}\biggr]\leq \frac{n_{0}(x_{0})-\frac{\lambda }{2}}{-1+t(1-\delta )(n_{0}(x_{0})-\frac{\lambda }{2})} \rightarrow -\infty \\& \quad \text{as } t\rightarrow \frac{1}{(1-\delta )(n_{0}(x_{0})-\frac{\lambda }{2})}. \end{aligned}$$

(3.19)

The proof of Theorem 1.4 is finished. □

Proof of Theorem 1.5

Differentiating the first equation in (1.4) with x, we acquire

$$\begin{aligned} \partial _{t}w_{x}+(2w_{x} -4w+\beta )w_{xx} =&2w_{x}^{2}-6w^{2}+P_{1}(D) \bigl[ w_{x}^{2} \bigr] \\ &{} +P_{2}(D)\bigl[2w_{x}^{2}+6w^{2} \bigr]- \lambda w_{x}. \end{aligned}$$

(3.20)

Making use of Remark 2.2 leads to

$$\begin{aligned} \biggl\vert \frac{d}{dt}w\bigl(t,p(t,x)\bigr) \biggr\vert =& \bigl\vert w_{t}+(2w_{x}-4w+\beta )w_{x} \bigr\vert \\ \lesssim &\Vert w_{0} \Vert _{H^{1}}^{2}+ \Vert w_{0} \Vert _{H^{1}} \end{aligned}$$

(3.21)

and

$$\begin{aligned} \biggl\vert \frac{d}{dt}w_{x}\bigl(t,p(t,x)\bigr) \biggr\vert =& \bigl\vert 2w_{x}^{2}-6w^{2}+P_{1}(D) \bigl[ w_{x}^{2} \bigr]+P_{2}(D) \bigl[2w_{x}^{2}+6w^{2} \bigr]- \lambda w_{x} \bigr\vert \\ \lesssim &\Vert w_{0} \Vert _{H^{1}}^{2}+ \Vert w_{0} \Vert _{H^{1}}. \end{aligned}$$

(3.22)

Eventually, we come to the identity

$$\begin{aligned}& \frac{d}{dt}n\bigl(t,p(t,x)\bigr) =2(n+2w_{x}-w)^{2}+2(w+w_{x})^{2}-2(2w_{x}-w)^{2}- \lambda n. \end{aligned}$$

(3.23)

That is,

$$\begin{aligned} \frac{d}{dt}\biggl[n+2w_{x}-w-\frac{\lambda }{4} \biggr]\bigl(t,p(t,x)\bigr) \geq& 2\biggl[n+2w_{x}-w-\frac{\lambda }{4} \biggr]^{2}\bigl(t,p(t,x)\bigr) \\ &{} -\bigl[C_{4} \Vert w_{0} \Vert _{H^{1}}^{2}+C_{5} \Vert w_{0} \Vert _{H^{1}}+C_{6}\bigr], \end{aligned}$$

where we use the inequality

$$\begin{aligned}& \biggl\vert - 2(w+w_{x})^{2}+2(2w_{x}-w)^{2}-2 \lambda w_{x}+\lambda w+ \frac{1}{8}\lambda ^{2} \biggr\vert \\& \quad \leq C_{4} \Vert w_{0} \Vert _{H^{1}}^{2}+C_{5} \Vert w_{0} \Vert _{H^{1}}+C_{6}. \end{aligned}$$

(3.24)

Setting

$$\begin{aligned}& \begin{gathered} n_{3}(t,x)=-\biggl[2\biggl(n+2w_{x}-w- \frac{\lambda }{4}\biggr) \bigl(t,p(t,x)\bigr)\biggr], \\ K_{1}=2\bigl( C_{4} \Vert w_{0} \Vert _{H^{1}}^{2}+C_{5} \Vert w_{0} \Vert _{H^{1}}+C_{6}\bigr) \end{gathered} \end{aligned}$$

(3.25)

gives rise to

$$\begin{aligned} \frac{dn_{3}(t)}{dt}\leq - n_{3}^{2}(t) + K_{1}. \end{aligned}$$

(3.26)

Let $\varepsilon \in (0,\frac{1}{2})$. Similar to the proof of Theorem 1.4, we choose certain $t_{0}\in (0,T)$ to satisfy $n_{3}(t_{0})<-\sqrt{ K_{1}+\frac{K_{1}}{\varepsilon }}$. Utilizing (3.26) gives rise to

$$ n_{3}(t)< -\sqrt{ K_{1}+\frac{K_{1}}{\varepsilon }}< -\sqrt{ \frac{K_{1}}{\varepsilon }} . $$

We check

$$\begin{aligned} 1-\varepsilon \leq \frac{ d }{ dt} \biggl( \frac{1}{n_{3}(t)} \biggr) \leq 1+\varepsilon . \end{aligned}$$

(3.27)

Applying $\lim_{t\rightarrow T^{-}}n_{3}(t)=-\infty$, $| w_{x}|\leq | w|\lesssim \| v_{0}\| _{H^{1}}$ and (3.24), we conclude

$$\begin{aligned} \lim_{t\rightarrow T^{-}}\Bigl[\sup_{x\in \mathbb{R}}(2w_{x}-w) (T-t)\Bigr]=0. \end{aligned}$$

Thus, we have

$$\begin{aligned} \lim_{t\rightarrow T^{-}}\biggl[\sup_{x\in \mathbb{R}}\biggl(n (t,x)- \frac{\lambda }{4}\biggr) (T-t)\biggr]= \frac{1}{2}, \end{aligned}$$

(3.28)

which finishes the proof of Theorem 1.5.

Proof of Theorem 1.6

Setting $M=\sup_{t\in [0,T]}\| v (t)\| _{H^{s} } >0$, $s>\frac{5}{2}$, we acquire $\| v_{xx}(t)\| _{L^{\infty }}\leq \| v(t) \| _{H^{s}}\leq M$. The function

$$\begin{aligned} \varphi _{N}(x)= \textstyle\begin{cases} 1, &x\leq 0, \\ e^{\theta x} ,& x\in (0,N), \\ e^{\theta N} ,& x\geq N \end{cases}\displaystyle \end{aligned}$$

satisfies $0\leq (\varphi _{N}(x))_{x}\leq \varphi _{N}(x)$, where $N\in \mathbb{N}^{\ast }$, $\theta \in (0,1)$. There exists a constant $M_{0}=M_{0}(\theta )>0$ such that

$$\begin{aligned} \varphi _{N}(x) \int _{\mathbb{R}}e^{-|x-y|}\frac{1}{\varphi _{N}(y)} \,dy \leq M_{0}. \end{aligned}$$

The first equation in (1.1) is written as

$$\begin{aligned} w_{t}+(-4w+\beta )w_{x} =\partial _{x}g \ast \bigl[2w_{x}^{2}+6w^{2}+ \partial _{x}\bigl(w_{x}^{2}\bigr)\bigr]-\lambda w. \end{aligned}$$

(4.1)

Then we acquire

$$\begin{aligned}& \frac{1}{2n}\,\frac{ d}{ dt} \Vert w \varphi _{N} \Vert _{L^{2n}}^{2n} \\& \quad =4 \int _{\mathbb{R}} \vert w \varphi _{N} \vert ^{2n}w_{x}\,dx-\beta \int _{ \mathbb{R}} \bigl[\partial _{x}(w \varphi _{N})-w (\varphi _{N})_{x} \bigr] (w \varphi _{N}) ^{2n-1} \,dx \\& \qquad {}+ \int _{\mathbb{R}} (w \varphi _{N}) ^{2n-1} \varphi _{N} \partial _{x}g \ast \bigl[2w_{x}^{2}+6w^{2}+ \partial _{x}\bigl(w_{x}^{2}\bigr)\bigr] \,dx- \lambda \int _{\mathbb{R}} (w \varphi _{N}) ^{2n } \,dx \\& \quad \leq 4 \Vert w_{x} \Vert _{L^{\infty }} \Vert w \varphi _{N} \Vert _{L^{2n}}^{2n}+\beta \Vert w \varphi _{N} \Vert _{L^{2n}}^{2n}- \lambda \Vert w \varphi _{N} \Vert _{L^{2n}}^{2n} \\& \qquad {}+ \bigl\Vert \varphi _{N} \partial _{x}g \ast \bigl[2w_{x}^{2}+6w^{2}+ \partial _{x}\bigl(w_{x}^{2}\bigr)\bigr] \bigr\Vert _{L^{2n}} \Vert w \varphi _{N} \Vert _{L^{2n}}^{2n-1}. \end{aligned}$$

(4.2)

Utilizing the Gronwall inequality and sending $n\rightarrow \infty$ in (4.2), we obtain

$$\begin{aligned} \Vert w \varphi _{N} \Vert _{L^{\infty }} \leq& e^{(4 M+\beta - \lambda )t} \biggl[ \Vert w_{0} \varphi _{N} \Vert _{L^{\infty }} \\ &{} + \int _{0}^{t} \bigl\Vert \varphi _{N} \partial _{x}g \ast \bigl[2w_{x}^{2}+6w^{2}+ \partial _{x}\bigl(w_{x}^{2}\bigr)\bigr] \bigr\Vert _{L^{ \infty }}\,d \tau \biggr] . \end{aligned}$$

(4.3)

Direct computation gives rise to

$$\begin{aligned} \Vert w \varphi _{N} \Vert _{L^{\infty }} \leq& e^{(4 M+\beta - \lambda )t} \biggl[ \Vert w_{0} \varphi _{N} \Vert _{L^{\infty }} +6M_{0}M \int _{0}^{t}\bigl( \Vert w\varphi _{N} \Vert _{L^{\infty }}+ \Vert w_{x}\varphi _{N} \Vert _{L^{ \infty }}\bigr) \,d \tau \biggr]. \end{aligned}$$

(4.4)

We arrive at

$$\begin{aligned} \Vert w _{x} \varphi _{N} \Vert _{L^{\infty }} \leq& e^{(6M+\beta -\lambda )t} \biggl[ \Vert w_{0,x} \varphi _{N} \Vert _{L^{\infty }}+(4M+6M_{0}M) \int _{0}^{t} \Vert w\varphi _{N} \Vert _{L^{\infty }}\,d \tau \\ &{} +\frac{5}{2}M_{0}M \int _{0}^{t} \Vert w_{x}\varphi _{N} \Vert _{L^{\infty }}\,d \tau \biggr] \end{aligned}$$

(4.5)

and

$$\begin{aligned} \Vert w _{xx} \varphi _{N} \Vert _{L^{\infty }} \leq& e^{(16M+\beta -\lambda )t} \biggl[ \Vert w_{0,x} \varphi _{N} \Vert _{L^{\infty }}+3M_{0}M \int _{0}^{t} \Vert w\varphi _{N} \Vert _{L^{\infty }}\,d \tau \\ & {}+ (12M+M_{0}M) \int _{0}^{t} \Vert w _{x} \varphi _{N} \Vert _{L^{\infty }} \,d \tau \\ & {}+M_{0}M \int _{0}^{t} \Vert w _{xx} \varphi _{N} \Vert _{L^{\infty }} \,d \tau \biggr]. \end{aligned}$$

(4.6)

Combining (4.4), (4.5) with (4.6), we achieve

$$\begin{aligned}& \Vert w\varphi _{N} \Vert _{L^{\infty }}+ \Vert w_{x} \varphi _{N} \Vert _{L^{\infty }}+ \Vert w_{xx}\varphi _{N} \Vert _{L^{\infty }} \\& \quad \leq C_{4}\bigl( \Vert w_{0}\varphi _{N} \Vert _{L^{\infty }}+ \Vert w_{0,x}\varphi _{N} \Vert _{L^{\infty }}+ \Vert w_{0,xx} \varphi _{N} \Vert _{L^{\infty }}\bigr) \\& \qquad{} +C_{4} \int _{0}^{t} \bigl( \Vert w\varphi _{N} \Vert _{L^{ \infty }} + \Vert w_{x}\varphi _{N} \Vert _{L^{\infty }}+ \Vert w_{xx}\varphi _{N} \Vert _{L^{\infty }}\bigr) \,d \tau , \end{aligned}$$

(4.7)

which leads to the estimate

$$\begin{aligned}& \sup_{t\in [0,T]}\bigl( \bigl\Vert e^{\theta x}w \bigr\Vert _{L^{\infty }}+ \bigl\Vert e^{\theta x}w_{x} \bigr\Vert _{L^{\infty }}+ \bigl\Vert e^{ \theta x}w_{xx} \bigr\Vert _{L^{\infty }}\bigr) \\& \quad \lesssim \bigl\Vert e^{\theta x}w_{0} \bigr\Vert _{L^{\infty }}+ \bigl\Vert e^{\theta x}w_{0,x} \bigr\Vert _{L^{\infty }}+ \bigl\Vert e^{ \theta x}w_{0,xx} \bigr\Vert _{L^{\infty }}. \end{aligned}$$

Thus, we acquire

$$\begin{aligned} \vert w \vert , \vert \partial _{x}w \vert , \bigl\vert \partial _{x}^{2}w \bigr\vert \thicksim O \bigl(e^{- \theta x}\bigr)\quad \text{as } x\rightarrow \infty \end{aligned}$$

uniformly on $[0,T]$.

Proof of Theorem 1.7

Let $s>0$. We give a scale of Banach spaces

$$\begin{aligned} E_{s}=\biggl\{ w\in C^{\infty }(\mathbb{R})\mid |\!|\!|w |\!|\!|_{s}=\sup_{k\in \mathbb{N}^{\ast }} \frac{s^{k} \Vert \partial _{x}^{k} w \Vert _{H^{2}}}{k! (k+1)^{-2}}< + \infty \biggr\} . \end{aligned}$$

Here, we denote $|\!|\!|\cdot |\!|\!|_{E_{s}}$ by $|\!|\!|\cdot |\!|\!|_{ s}$ for simplicity. $E_{s}$ is continuously embedded in $E_{s'}$ with $0< s'< s$ and $|\!|\!|w |\!|\!|_{s'}\leq |\!|\!|w |\!|\!|_{s}$. A function w in $E_{s}$ is a real analytic function on $\mathbb{R}$.

We present several related lemmas.

Lemma 5.1

([12])

Assume $s>0$. Then, for all $u, v\in E_{s}$, it holds that

$$\begin{aligned} |\!|\!|uv|\!|\!|_{s}\leq C |\!|\!|u |\!|\!|_{s} |\!|\!|v |\!|\!|_{s}, \end{aligned}$$

where $C >0$ is independent of s.

Lemma 5.2

([12])

There exists a positive constant C, for all $0 < s'< s\leq 1$, such that

$$\begin{aligned}& |\!|\!|\partial _{x} u|\!|\!|_{s'}\leq \frac{C}{s-s'} |\!|\!|u |\!|\!|_{s}, \\& \bigl|\!\bigl|\!\bigl|P_{1}(D) u\bigr|\!\bigr|\!\bigr|_{s'}\leq |\!|\!|u |\!|\!|_{s},\qquad \bigl|\!\bigl|\!\bigl|P_{2}(D) u\bigr|\!\bigr|\!\bigr|_{s'} \leq |\!|\!|u |\!|\!|_{s}. \end{aligned}$$

Lemma 5.3

([12])

Let $\{X_{s} \}_{0< s<1}$ be a scale of decreasing Banach spaces. $X_{s}\hookrightarrow X_{s'}$ for all $s'< s$. T, R, and C are positive constants. Consider the Cauchy problem

$$\begin{aligned} \frac{du}{dt}= F\bigl(t,u(t)\bigr),\quad u(0) = 0. \end{aligned}$$

(5.1)

$F(t,u)$ satisfies the following conditions:

(1) Let $0 < s'< s<1$. $u(t)$ is holomorphic for $|t| < T$ and continuous on $|t| < T$ with values in $X_{s}$. $u(t)$ satisfies $\sup_{ |t |< T}|\!|\!|u(t)|\!|\!|_{s} < R$. Then $t \rightarrow F(t, u(t))$ is holomorphic on $|t| < T$ with values in $X_{s'}$.

(2) For $0 < s'< s\leq 1$ and $u, v\in X_{s}$ with $|\!|\!|u|\!|\!|_{s}< R$ and $|\!|\!|v|\!|\!|_{s}< R$, it holds that

$$\begin{aligned} \sup_{|t|\leq T} \bigl|\!\bigl|\!\bigl|F(t,u)-F(t,v)\bigr|\!\bigr|\!\bigr|_{s'} \leq& \frac{C}{s-s'} |\!|\!|u-v |\!|\!|_{s}. \end{aligned}$$

(3) Let $T_{0}\in (0, T)$. There exists $M>0$, for all $0< s<1$, such that

$$\begin{aligned} \sup_{ |t |< T}\bigl|\!\bigl|\!\bigl|F(t,0)\bigr|\!\bigr|\!\bigr|_{s} < \frac{M}{1-s}. \end{aligned}$$

Then problem (5.1) admits a unique solution $u(t)$ which is holomorphic for $|t| <(1 -s)T_{0}$ with values in $X_{s}$ for all $s\in (0, 1)$.

Let $u_{1}=w$, $u_{2}=w_{x}$. The pair $(u_{1}, u_{2})$ satisfies the problem

$$ \textstyle\begin{cases} u_{1,t}=4u_{1}u_{2}-u_{2}^{2}+F_{1}(u_{1},u_{2}), \\ u_{2,t}=4\partial _{x} (u_{1}u_{2})-\partial _{x}(u_{2}^{2})+F_{2}(u_{1},u_{2}), \\ u_{1}(0,x)=u_{10}(x)=w_{ 0}(x), \\ u_{2}(0,x)=u_{20}(x)=w _{0,x}(x), \end{cases} $$

(5.2)

where

$$\begin{aligned}& F_{1}(u_{1},u_{2})=P_{1}(D) \bigl[2u_{2}^{2}+6u_{1}^{2} \bigr] +P_{2}(D)\bigl[u_{2}^{2} \bigr]- \lambda u_{1}-\beta u_{2}, \\& F_{2}(u_{1},u_{2})=\partial _{x}P_{1}(D)\bigl[2u_{2}^{2}+6u_{1}^{2} \bigr] + \partial _{x}P_{2}(D)\bigl[u_{2}^{2} \bigr]- \lambda \partial _{x} ( u_{1}) - \beta \partial _{x} ( u_{2}). \end{aligned}$$

Proof of Theorem 1.7

We acquire that $F_{1}(u_{1}, u_{2})$ and $F_{2}(u_{1}, u_{2})$ do not depend on t explicitly. We only need to verify conditions (1) and (2) in Lemma 5.3 for $F_{1}(u_{1}, u_{2})$ and $F_{2}(u_{1}, u_{2})$. Making use of Lemmas 5.1 and 5.2 gives rise to

$$\begin{aligned}& \begin{aligned}[b] \bigl|\!\bigl|\!\bigl|F_{1}(u_{1},u_{2}) \bigr|\!\bigr|\!\bigr|_{s'} \leq{}& C |\!|\!|u_{1}|\!|\!|_{s} |\!|\!|u_{2}|\!|\!|_{s} + |\!|\!|u_{2} |\!|\!|^{2}_{s}+ \frac{C}{s - s'} \bigl(2|\!|\!|u_{2}|\!|\!|^{2}_{s}+6|\!|\!|u_{1} |\!|\!|^{2}_{s}\bigr) \\ &{} +\lambda |\!|\!|u_{1}|\!|\!|_{s}+\beta |\!|\!|u_{2}|\!|\!|_{s}, \end{aligned} \end{aligned}$$

(5.3)

$$\begin{aligned}& \begin{aligned}[b] \bigl|\!\bigl|\!\bigl|F_{2}(u_{1},u_{2}) \bigr|\!\bigr|\!\bigr|_{s'}\leq{}& \frac{C}{s - s'} |\!|\!|u_{1} |\!|\!|_{s} |\!|\!|u_{2}|\!|\!|_{s} + \frac{C}{s - s'} |\!|\!|u_{2}|\!|\!|^{2}_{s} \\ &{} + \frac{C}{s - s'} \bigl(2|\!|\!|u_{2} |\!|\!|^{2}_{s}+6|\!|\!|u_{1} |\!|\!|^{2}_{s}\bigr) +\frac{C}{s - s'}\lambda |\!|\!|u_{1}|\!|\!|_{s}+\frac{C}{s - s'}\beta |\!|\!|u_{2}|\!|\!|_{s}, \end{aligned} \end{aligned}$$

(5.4)

where C is a positive constant. Then condition (1) in Lemma 5.3 holds.

In order to verify condition (2) in Lemma 5.3, we obtain

$$\begin{aligned} \begin{aligned}[b] &\bigl|\!\bigl|\!\bigl|F_{1}(u_{1}, u_{2}) - F_{1}( \bar{ u}_{1}, \bar{ u}_{2})\bigr|\!\bigr|\!\bigr|_{s'} \\ &\quad \leq \bigl|\!\bigl|\!\bigl|F_{1}(u_{1}, u_{2}) - F_{1}( \bar{ u}_{1}, u_{2} )\bigr|\!\bigr|\!\bigr|_{s'} +\bigl|\!\bigl|\!\bigl|F_{1}( \bar{u}_{1}, u_{2}) - F_{1}( \bar{ u}_{1}, \bar{ u}_{2})\bigr|\!\bigr|\!\bigr|_{s'}, \end{aligned} \end{aligned}$$

(5.5)

$$\begin{aligned} \begin{aligned}[b] & \bigl|\!\bigl|\!\bigl|F_{2}(u_{1}, u_{2}) - F_{2}( \bar{ u}_{1}, \bar{ u}_{2})\bigr|\!\bigr|\!\bigr|_{s'} \\ &\quad \leq \bigl|\!\bigl|\!\bigl|F_{2}(u_{1}, u_{2}) - F_{2}( \bar{ u}_{1}, u_{2} )\bigr|\!\bigr|\!\bigr|_{s'} +\bigl|\!\bigl|\!\bigl|F_{2}( \bar{u}_{1}, u_{2}) - F_{2}( \bar{ u}_{1}, \bar{ u}_{2})\bigr|\!\bigr|\!\bigr|_{s'} . \end{aligned} \end{aligned}$$

(5.6)

Taking advantage of Lemmas 5.1, 5.2 and the assumptions $|\!|\!|u_{1}|\!|\!|_{s}\leq |\!|\!|u_{10}|\!|\!|_{s}+R$ and $|\!|\!|u_{2}|\!|\!|_{s}\leq |\!|\!|u_{20}|\!|\!|_{s}+R$ yields

$$\begin{aligned}& \bigl|\!\bigl|\!\bigl|F_{1}(u_{1}, u_{2}) - F_{1}( \bar{u}_{1}, u_{2} )\bigr|\!\bigr|\!\bigr|_{s'} \\& \quad \leq C |\!|\!|u_{1}-\bar{ u}_{1}|\!|\!|_{s} |\!|\!|u_{2}|\!|\!|_{s} + \frac{C}{s - s'} \bigl|\!\bigl|\!\bigl|u_{1}^{2}-\bar{ u}_{1}^{2}\bigr|\!\bigr|\!\bigr|_{s} +\lambda |\!|\!|u_{1}- \bar{ u}_{1}|\!|\!|_{s} \\& \quad \leq C (|\!|\!|u_{20}|\!|\!|_{s}+R) |\!|\!|u_{1}-\bar{ u}_{1}|\!|\!|_{s} +C(|\!|\!|u_{10}|\!|\!|_{s}+R+ \lambda ) |\!|\!|u_{1}- \bar{ u}_{1}|\!|\!|_{s} , \end{aligned}$$

(5.7)

$$\begin{aligned}& \bigl|\!\bigl|\!\bigl|F_{1}(\bar{u}_{1}, u_{2}) - F_{1}( \bar{ u}_{1}, \bar{ u}_{2})\bigr|\!\bigr|\!\bigr|_{s'} \\& \quad \leq |\!|\!|\bar{u}_{1}|\!|\!|_{s}|\!|\!|u_{2}-\bar{ u}_{2}|\!|\!|_{s} + \bigl|\!\bigl|\!\bigl|u_{2}^{2}- \bar{ u}_{2}^{2}\bigr|\!\bigr|\!\bigr|_{s}+ \frac{C}{s - s'} \bigl|\!\bigl|\!\bigl|u_{2}^{2}- \bar{ u}_{2}^{2}\bigr|\!\bigr|\!\bigr|_{s} +\beta |\!|\!|u_{2}-\bar{ u}_{2}|\!|\!|_{s} \\& \quad \leq C (|\!|\!|u_{10}|\!|\!|_{s}+R) |\!|\!|u_{1}-\bar{ u}_{1}|\!|\!|_{s} +C(|\!|\!|u_{20}|\!|\!|_{s}+R+ \beta ) |\!|\!|u_{1}- \bar{ u}_{1}|\!|\!|_{s} , \end{aligned}$$

(5.8)

$$\begin{aligned}& \bigl|\!\bigl|\!\bigl|F_{2}(u_{1}, u_{2}) - F_{2}( \bar{ u}_{1}, u_{2} )\bigr|\!\bigr|\!\bigr|_{s'} \\& \quad \leq \frac{C}{s - s'} |\!|\!|u_{1}-\bar{ u}_{1} |\!|\!|_{s} |\!|\!|u_{2}|\!|\!|_{s} + \frac{C}{s - s'} \bigl|\!\bigl|\!\bigl|u_{1}^{2}-\bar{ u}_{1}^{2}\bigr|\!\bigr|\!\bigr|_{s} + \frac{C}{s - s'} \lambda |\!|\!|u_{1}-\bar{ u}_{1}|\!|\!|_{s} \\& \quad \leq C (|\!|\!|u_{20}|\!|\!|_{s}+R) |\!|\!|u_{1}-\bar{ u}_{1}|\!|\!|_{s} +C(|\!|\!|u_{10}|\!|\!|_{s}+R+ \lambda ) |\!|\!|u_{1}- \bar{ u}_{1}|\!|\!|_{s} , \end{aligned}$$

(5.9)

$$\begin{aligned}& \bigl|\!\bigl|\!\bigl|F_{2}(\bar{u}_{1}, u_{2}) - F_{2}( \bar{ u}_{1}, \bar{ u}_{2})\bigr|\!\bigr|\!\bigr|_{s'} \\& \quad \leq \frac{C}{s - s'}|\!|\!|\bar{u}_{1}|\!|\!|_{s}|\!|\!|u_{2}-\bar{ u}_{2}|\!|\!|_{s} +\frac{C}{s - s'} \bigl|\!\bigl|\!\bigl|u_{2}^{2}- \bar{ u}_{2}^{2}\bigr|\!\bigr|\!\bigr|_{s} \\& \qquad {}+ \frac{C}{s - s'} \bigl|\!\bigl|\!\bigl|u_{1}^{2}-\bar{ u}_{1}^{2}\bigr|\!\bigr|\!\bigr|_{s} + \frac{C}{s - s'} \beta |\!|\!|u_{2}-\bar{ u}_{2}|\!|\!|_{s} \\& \quad \leq C (|\!|\!|u_{10}|\!|\!|_{s}+R) |\!|\!|u_{2}-\bar{ u}_{2}|\!|\!|_{s} +(|\!|\!|u_{20}|\!|\!|_{s}+R+ \beta ) |\!|\!|u_{2}- \bar{ u}_{2}|\!|\!|_{s} . \end{aligned}$$

(5.10)

From (5.5)–(5.10), we check that condition (2) in Lemma 5.3 holds. Replacing $s'$ with s and s with 1 and applying condition (2) in Lemma 5.3 give rise to that condition (3) in Lemma 5.3 holds. This finishes the proof of Theorem 1.7. □