2.1 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.
2.2 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)