Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space

Abstract

In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space $B^{\frac{1}{2}}_{2,1}$. Next we derive a new blow-up criterion for strong solutions to the equation. Then we give a global existence result for strong solutions to the equation. Finally, we present two new blow-up results and the exact blow-up rate for strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.

1 Introduction

In this paper we consider the Cauchy problem for the following generalized Camassa–Holm equation,

$$\begin{aligned} \left\{ \begin{array}{ll} u_t-u_{txx}=\frac{1}{2}(3u_{x}^2-2u_{x}u_{xxx}-u_{xx}^2),~~~~ t>0,\\ u(0,x)=u_{0}(x), \end{array} \right. \end{aligned}$$

(1.1)

which can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{llll} m=u-u_{xx},\\ m_t-u_xm_x=-\frac{1}{2}m^2+um+\frac{1}{2}u_{x}^2-\frac{1}{2}u^2 , ~~ t>0,\\ m(0,x)=u(0,x)-u_{xx}(0,x)=m_0(x). \end{array} \right. \end{aligned}$$

(1.2)

The Eq. (1.1) was proposed recently by Novikov in [38]. He showed that the equation (1.1) is integrable by using as definition of integrability the existence of an infinite hierarchy of quasi-local higher symmetries [38] and it belongs to the following class [38]:

$$\begin{aligned} (1-\partial ^2_x)u_t=F(u,u_x,u_{xx},u_{xxx}), \end{aligned}$$

(1.3)

which has attracted much interest, particularly in the possible integrable members of (1.3).

The most celebrated integrable members of (1.3) which have quadratic nonlinearity are the well-known Camassa–Holm (CH) equation [4] and the famous Degasperis-Procesi (DP) equation [23]:

$$\begin{aligned} (1-\partial ^2_x)u_t&=3uu_x-2u_{x}u_{xx}-uu_{xxx}, \end{aligned}$$

(1.4)

$$\begin{aligned} (1-\partial ^2_x)u_t&=4uu_x-3u_{x}u_{xx}-uu_{xxx}. \end{aligned}$$

(1.5)

Both the CH equation and the DP equation can be regarded as a shallow water wave equation [4, 16, 24]. They are completely integrable. That means that the system can be transformed into a linear flow at constant speed in suitable action-angle variables (in the sense of infinite-dimensional Hamiltonian systems), for a large class of initial data [4, 8, 17, 22]. They also have a bi-Hamiltonian structure [7, 22, 27], and admit exact peaked solitons, which are orbitally stable [19]. It is worth mentioning that the peaked solitons present the characteristic for the traveling water waves of greatest height and largest amplitude and arise as solutions to the free-boundary problem for incompressible Euler equations over a flat bed, cf. [5, 10, 14, 15, 40]. The main difference between DP equation and CH equation is that DP equation has short waves [36] and the periodic shock waves [26].

The local well-posedness and ill-posedness for the Cauchy problem of the CH equation in Sobolev spaces and Besov spaces were discussed in [11, 12, 20, 29, 35, 39]. It was shown that there exist global strong solutions to the CH equation [9, 11, 12] and finite time blow-up strong solutions to the CH equation [9, 11,12,13]. The existence and uniqueness of global weak solutions to the CH equation were proved in [18, 44]. The global conservative and dissipative solutions of CH equation were investigated in [2, 3].

The local well-posedness of the Cauchy problem of the DP equation in Sobolev spaces and Besov spaces was established in [28, 30, 47]. Similar to the CH equation, the DP equation has also global strong solutions [33, 48, 50] and finite time blow-up solutions [25, 26, 33, 34, 47,48,49,50]. It also has global weak solutions [6, 25, 49, 50].

The third celebrated integrable member of (1.3) which has cubic nonlinearity is the known Novikov equation [38]:

$$\begin{aligned} (1-\partial ^2_x)u_t=3uu_{x}u_{xx}+u^2u_{xxx}-4u^2u_x. \end{aligned}$$

(1.6)

It was showed that the Novikov equation is integrable, possesses a bi-Hamiltonian structure, and admits exact peakon solutions $u(t,x)=\pm \sqrt{c}e^{|x-ct|}$ with $c>0$ [31].

The local well-posedness for the Novikov equation in Sobolev spaces and Besov spaces was studied in [42, 43, 45, 46]. The global existence of strong solutions under some sign conditions were established in [42] and the blow-up phenomena of the strong solutions were shown in [46]. The global weak solutions for the Novikov equation were studied in [41].

Recently, the Cauchy problem of (1.1) in the Besov spaces $B^{s}_{p,r},~s>max\{\frac{1}{p},\frac{1}{2}\}$ has been studied in [32]. To our best knowledge, the Cauchy problem of (1.1) in the critical Besov space $B^{\frac{1}{2}}_{2,1}$ has not been studied yet. In this paper we first investigate the local well-posedness of (1.2) with initial data in the critical Besov space $B^{\frac{1}{2}}_{2,1}$. The main idea is based on the Littlewood–Paley theory, transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma. Then, we prove a new blow-up criteria by the energy method, which is more precise than the blow-up criteria derived in [32]. By virtue of a conservation law, we obtain two new blow-up results. Finally, we conclude the exact blow-up rate of blowing-up solutions m(t, x) to (1.1).

The paper is organized as follows. In Sect. 2 we introduce some preliminaries which will be used in sequel. In Sect. 3 we prove the local well-posedness of (1.1) by using Littlewood–Paley and transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma. In Sect. 4, we derive a conservation law and a blow-up criterion. In Sect. 5, we show the global existence of strong solution to (1.1). Section 6 is devoted to the study of blow-up phenomena of (1.1). We present two blow-up results and the exact blow-up rate of blowing-up solutions to (1.1).

2 Preliminaries

In this section, we first recall the Littlewood–Paley decomposition and Besov spaces (for more details to see [1]). Let ${\mathcal {C}}$ be the annulus $\{\xi \in {\mathbb {R}}^{d}\big |\frac{3}{4}\le |\xi |\le \frac{8}{3}\}.$ There exist radial functions $\chi$ and $\varphi$, valued in the interval [0, 1], belonging respectively to ${\mathcal {D}}(B(0,\frac{4}{3}))$ and ${\mathcal {D}}({\mathcal {C}})$, and such that

$$\begin{aligned}&\forall \xi \in {\mathbb {R}}^{d},~\chi (\xi )+\sum _{j\ge 0}\varphi (2^{-j}\xi )=1, \\&|j-j'|\ge 2\Rightarrow Supp ~\varphi (2^{-j}\xi )\cap Supp ~\varphi (2^{-j'}\xi )=\emptyset , \\&j\ge 1\Rightarrow Supp ~\chi (\xi )\cap Supp ~\varphi (2^{-j'}\xi )=\emptyset . \end{aligned}$$

Define the set $\widetilde{{\mathcal {C}}}=B(0,\frac{2}{3})+{\mathcal {C}}$. Then we have

$$\begin{aligned} |j-j'|\ge 5\Rightarrow 2^{j'}\widetilde{{\mathcal {C}}}\cap 2^{j}{\mathcal {C}}=\emptyset . \end{aligned}$$

Further, we have

$$\begin{aligned} \forall \xi \in {\mathbb {R}}^{d},~\frac{1}{2}\le \chi ^{2}(\xi )+\sum _{j\ge 0}\varphi ^{2}(2^{-j}\xi )\le 1. \end{aligned}$$

Denote ${\mathcal {F}}$ by the Fourier transform and ${\mathcal {F}}^{-1}$ by its inverse. From now on, we write $h={\mathcal {F}}^{-1}\varphi$ and ${\widetilde{h}}={\mathcal {F}}^{-1}\chi$. The nonhomogeneous dyadic blocks $\Delta _{j}$ are defined by

$$\begin{aligned}&\Delta _{j}u=0\,\,\, if\,\,\, j\le -2, \quad \Delta _{-1}u=\chi (D)u=\int _{{\mathbb {R}}^{d}}{\widetilde{h}}(y)u(x-y)dy, \\&and,\quad \Delta _{j}u=\varphi (2^{-j}D)u=2^{jd}\int _{{\mathbb {R}}^{d}}h(2^{j}y)u(x-y)dy ~~~if~~ j\ge 0, \\&S_{j}u=\sum _{j'\le j-1}\Delta _{j'}u. \end{aligned}$$

The nonhomogeneous Besov spaces are denoted by $B^{s}_{p,r}({\mathbb {R}}^d)$

$$\begin{aligned} B^{s}_{p,r}=\left\{ u\in S'\big | \Vert u\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}=\left( \sum _{j\ge -1}2^{rjs}\Vert \Delta _{j}u\Vert ^{r}_{L^{p}({\mathbb {R}}^d)}\right) ^{\frac{1}{r}}<\infty \right\} . \end{aligned}$$

Next we introduce some useful lemmas and propositions about Besov spaces which will be used in the sequel.

Proposition 2.1

[1] Let $1\le p_{1} \le p_{2} \le \infty$ and $1\le r_{1} \le r_{2} \le \infty$, and let s be a real number. Then we have

$$\begin{aligned} B^{s}_{p_{1},r_{1}}({\mathbb {R}}^d)\hookrightarrow B^{s-d(\frac{1}{p_{1}}-\frac{1}{p_{2}})}_{p_{2},r_{2}}({\mathbb {R}}^d). \end{aligned}$$

If $s>\frac{d}{p}~or ~s=\frac{d}{p},~r=1$, we then have

$$\begin{aligned} B^{s}_{p,r}({\mathbb {R}}^d)\hookrightarrow L^{\infty }({\mathbb {R}}^d). \end{aligned}$$

Lemma 2.2

[1] A constant C exists which satisfies the following properties. If $s_1$ and $s_2$ are real numbers such that $s_1<s_2$ and $\theta \in (0, 1)$, then we have, for any $(p,r)\in [1,\infty ]^2$ and $u\in {\mathcal {S}}_{h}',$

$$\begin{aligned}&\Vert u\Vert _{B_{p,r}^{\theta s_1+(1-\theta )s_2}}\le \Vert u\Vert ^{\theta }_{B_{p,r}^{s_1}} \Vert u\Vert ^{1-\theta }_{B_{p,r}^{s_2}} \quad and \end{aligned}$$

(2.1)

$$\begin{aligned}&\Vert u\Vert _{B_{p,1}^{\theta s_1+(1-\theta )s_2}}\le \frac{C}{s_2-s_1}\left( \frac{1}{\theta }+\frac{1}{1-\theta }\right) \Vert u\Vert ^{\theta }_{B_{p,\infty }^{s_1}} \Vert u\Vert ^{1-\theta }_{B_{p,\infty }^{s_2}}. \end{aligned}$$

(2.2)

Lemma 2.3

[1] For any positive real number s and any (p, r) in $[1,\infty ]^{2}$, the space $L^{\infty }({\mathbb {R}}^d)\cap B^{s}_{p,r}({\mathbb {R}}^d)$ is an algebra, and a constant C exists such that

$$\begin{aligned} \Vert uv\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}\le C\left( \Vert u\Vert _{L^{\infty }({\mathbb {R}}^d)}\Vert v\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}+\Vert u\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}\Vert v\Vert _{L^{\infty }({\mathbb {R}}^d)}\right) . \end{aligned}$$

If $s>\frac{d}{p}$ or $s=\frac{d}{p},~r=1$, then we have

$$\begin{aligned} \Vert uv\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}\le C\Vert u\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}\Vert v\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}. \end{aligned}$$

The following two lemmas is crucial to study well-posedness in the critical space $B^{\frac{1}{2}}_{2,1}({\mathbb {R}})$.

Lemma 2.4

[37] For any $a\in B^{-\frac{1}{2}}_{2,\infty }({\mathbb {R}})$ and $b\in B^{\frac{1}{2}}_{2,1}({\mathbb {R}})$, there exists a constant C such that

$$\begin{aligned} \Vert ab\Vert _{B^{-\frac{1}{2}}_{2,\infty }({\mathbb {R}})}\le C\Vert a\Vert _{B^{-\frac{1}{2}}_{2,\infty }({\mathbb {R}})}\Vert b\Vert _{B^{\frac{1}{2}}_{2,1}({\mathbb {R}})}. \end{aligned}$$

(2.3)

Lemma 2.5

(Morse-type estimate, [1, 20]) Let $s>\max \{\frac{d}{p},\frac{d}{2}\}$ and (p, r) in $[1,\infty ]^{2}$ or $s=\frac{d}{2},~ p=2, ~r=1$. For any $a\in B^{s-1}_{p,r}({\mathbb {R}}^d)$ and $b\in B^{s}_{p,r}({\mathbb {R}}^d)$, there exists a constant C such that

$$\begin{aligned} \Vert ab\Vert _{B^{s-1}_{p,r}({\mathbb {R}}^d)}\le C\Vert a\Vert _{B^{s-1}_{p,r}({\mathbb {R}}^d)}\Vert b\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}. \end{aligned}$$

Lemma 2.6

[21] For any $s\in {\mathbb {R}}$, $\varepsilon \in (0,1]$ and $f\in B^{s+\varepsilon }_{2,1}({\mathbb {R}})$, there exists a constant C such that

$$\begin{aligned} \Vert f\Vert _{B^{s}_{2,1}({\mathbb {R}})}\le \frac{C}{\varepsilon }\Vert f\Vert _{B^{s}_{2,\infty }({\mathbb {R}})} \ln \bigg (e+\frac{\Vert f\Vert _{B^{s+\varepsilon }_{2,1}({\mathbb {R}})}}{\Vert f\Vert _{B^{s}_{2,\infty }({\mathbb {R}})}}\bigg ).\end{aligned}$$

Remark 2.7

[1] Let $s\in {\mathbb {R}},1\le p,r\le \infty$. Then the following properties hold true:

• (i) $B^s_{p,r}({\mathbb {R}}^d)$ is a Banach space and continuously embedding into ${\mathcal {S}}'({\mathbb {R}}^d)$, where ${\mathcal {S}}'({\mathbb {R}}^d)$ is the dual space of the Schwartz space ${\mathcal {S}}({\mathbb {R}}^d)$.

• (ii) If $p,r<\infty$, then ${\mathcal {S}}({\mathbb {R}}^d)$ is dense in $B^s_{p,r}({\mathbb {R}}^d)$.

• (iii) If $u_n$ is a bounded sequence of $B^s_{p,r}({\mathbb {R}}^d)$, then an element $u\in B^s_{p,r}({\mathbb {R}}^d)$ and a subsequence $u_{n_k}$ exist such that

  $$\begin{aligned} \lim _{k\rightarrow \infty }u_{n_k}=u~~in~~{\mathcal {S}}'({\mathbb {R}}^d)~~and~~\Vert u\Vert _{B^s_{p,r}({\mathbb {R}}^d)}\le C\liminf _{k\rightarrow \infty }\Vert u_{n_k}\Vert _{B^s_{p,r}({\mathbb {R}}^d)}. \end{aligned}$$

• (iv) $B^s_{2,2}({\mathbb {R}}^d)=H^s({\mathbb {R}}^d)$.

The following Osgood’s lemma appears as a substitute for Gronwall’s lemma.

Lemma 2.8

(Osgood’s lemma, [1]) Let $\rho \ge 0$ be a measurable function, $\gamma >0$ be a locally integrable function and $\mu$ be a continuous and increasing function. Assume that, for some nonnegative real number c, the function $\rho$ satisfies

$$\begin{aligned} \rho (t)\le c+\int ^t_{t_0}\gamma (t')\mu (\rho (t'))dt'. \end{aligned}$$

If $c>0$, then $-{\mathcal {M}}(\rho (t))+{\mathcal {M}}(c)\le \displaystyle \int ^t_{t_0}\gamma (t')dt'$ with ${\mathcal {M}}(x)=\displaystyle \int ^1_x\frac{dr}{\mu (r)}$.

If $c=0$ and $\mu$ verifies the condition $\displaystyle \int ^1_0\frac{dr}{\mu (r)}=+\infty$, then the function $\rho =0$.

Remark 2.9

In this paper, we set $\mu (r)=r(1-\ln r)$ which satisfies the condition $\displaystyle \int ^1_0\frac{dr}{\mu (r)}=+\infty$. A simple calculation shows that ${\mathcal {M}}(x)=\ln (1-\ln x)$, we deduce that

$$\begin{aligned} \rho (t)\le ec^{\exp {\displaystyle \int ^t_{t_0}-\gamma (t')dt'}},~~~if~~c>0. \end{aligned}$$

Now we introduce a priori estimates for the following transport equation.

$$\begin{aligned} \left\{ \begin{array}{ll} f_{t}+v\nabla f=g,\\ f|_{t=0}=f_{0}. \end{array} \right. \end{aligned}$$

(2.4)

Lemma 2.10

(A priori estimates in Besov spaces, [1, 20, 21]) Let $1\le p \le p_1\le \infty$, $1\le r\le \infty$, $s\ge -d\min (\frac{1}{p_1},\frac{1}{p'})$. For the solution $f\in L^{\infty }([0,T];B^s_{p,r}({\mathbb {R}}^d))$ of (2.4) with velocity $\nabla v\in L^1([0,T];B^s_{p,r}({\mathbb {R}}^d)\cap L^{\infty }({\mathbb {R}}^d))$, initial data $f_0\in B^s_{p,r}({\mathbb {R}}^d)$ and $g\in L^1([0,T];B^s_{p,r}({\mathbb {R}}^d))$, we have the following statements. If $s\ne 1+{1\over p}$ or $r=1$,

$$\begin{aligned}&\Vert f(t)\Vert _{B^{s}_{p,r}({\mathbb {R}}^d)}\nonumber \\&\quad \le \Vert f_0\Vert _{B^s_{p,r}({\mathbb {R}}^d)}+\int ^t_0\bigg (\Vert g(t')\Vert _{B^s_{p,r}({\mathbb {R}}^d)} +CV'_{p_1}(t')\Vert f(t')\Vert _{B^s_{p,r}({\mathbb {R}}^d)}\bigg )dt', \end{aligned}$$

(2.5)

$$\begin{aligned}&\Vert f\Vert _{(B^{s}_{p,r}({\mathbb {R}}^d))}\nonumber \\&\quad \le \bigg (\Vert f_0\Vert _{B^s_{p,r}({\mathbb {R}}^d)}+\int ^t_0\exp (-CV_{p_1}(t')) \Vert g(t')\Vert _{B^s_{p,r}({\mathbb {R}}^d)}dt'\bigg )\exp (CV_{p_1}(t)), \end{aligned}$$

(2.6)

where $V_{p_1}(t)=\displaystyle \int ^t_0\Vert \nabla v\Vert _{B^{\frac{d}{p_1}}_{p_1,\infty }({\mathbb {R}}^d)\cap L^{\infty }({\mathbb {R}}^d)}dt'$, if $s<1+\frac{d}{p_1}$; $V_{p_1}(t)=\displaystyle \int ^t_0\Vert \nabla v\Vert _{B^{s-1}_{p_1,r}({\mathbb {R}}^d)}dt'$, if $s>1+\frac{d}{p_1}$ or $s=1+\frac{d}{p_1}, r=1$, and C is a constant depending only on $s,~p,~p_1$ and r.

Lemma 2.11

[1] Let s be as in the statement of Lemma 2.10. Let $f_0 \in B^s_{p,r}({\mathbb {R}}^d)$, $g \in L^1([0, T];B^s_{p,r}({\mathbb {R}}^d))$, and v be a time-dependent vector field such that $v\in L^\rho ([0, T];B^{-M}_{\infty ,\infty }({\mathbb {R}}^d))$ for some $\rho > 1$ and $M >0$, and

$$\begin{aligned}&\nabla v\in L^1([0, T];B_{p_1,\infty }^{\frac{d}{p}}({\mathbb {R}}^d) ), ~~if ~s<1+\frac{d}{p_1}, \\&\nabla v\in L^1([0, T];B_{p_1,\infty }^{s-1}({\mathbb {R}}^d)), ~~if~ s>1+\frac{d}{p_1} ~~or~~s=1+\frac{d}{p_1}~ and~ r=1 . \end{aligned}$$

Then, (2.4) has a unique solution f in

• the space ${\mathcal {C}}([0,T];B_{p,r}^s({\mathbb {R}}^d)), if~ r<\infty ,$

• the space $(\bigcap _{s'<s}{\mathcal {C}}([0,T];B_{p,\infty }^{s'} ({\mathbb {R}}^d)))\bigcap {\mathcal {C}}_{w}([0,T];B_{p,\infty }^{s}({\mathbb {R}}^d))),~~if~ r=\infty .$ Moreover, the inequalities of Lemma 2.10 hold true.

Lemma 2.12

[35] Let $s>\frac{1}{p}, r<\infty \ (or\ s=\frac{1}{p}, 1\le p<\infty , r=1), k\in {\mathbb {N}}$ and a constant C depending only on $s, p, r, v, \ and\ g$. If (2.4) satisfies the following conditions for all $f, {\widetilde{f}}\in B^{s}_{p,r}$,

$$\begin{aligned}&(1)\Vert v\Vert _{B^{s+1}_{p,r}}\le C(1+\Vert f\Vert ^{k}_{B^{s}_{p,r}}), \\&(2)\Vert g\Vert _{B^{s}_{p,r}}\le C(1+\Vert f\Vert ^{k}_{B^{s}_{p,r}}), \\&(3)\Vert v(f)-v({\widetilde{f}})\Vert _{B^{s}_{p,r}}\le C\Vert f-{\widetilde{f}}\Vert _{B^{s}_{p,r}}(1+\Vert f\Vert ^{k-1}_{B^{s}_{p,r}}), \\&(4)\Vert g(f)-g({\widetilde{f}})\Vert _{B^{s}_{p,r}}\le C\Vert f-{\widetilde{f}}\Vert _{B^{s}_{p,r}}(1+\Vert f\Vert ^{k}_{B^{s}_{p,r}}). \end{aligned}$$

Denote $n \in {\mathbb {N}}.$ If $u^n_0$ tends to $u_0 \in B^{s}_{p,r}$ and $u^n$ tends to $u \in C([0, T]; B^{s-1}_{p,r})$, then $u^n$ tends to $u \in C([0, T]; B^{s}_{p,r})$.

Notations Since all space of functions in the following sections are over ${\mathbb {R}},$ for simplicity, we drop ${\mathbb {R}}$ in our notations of function spaces if there is no ambiguity.

3 Local well-posedness

In this section, we establish local well-posedness of (1.2) in the critical Besov space $B^{\frac{1}{2}}_{2,1}$. Our main result can be stated as follows.

Theorem 3.1

Let $~m_0\in B^{\frac{1}{2}}_{2,1}.$ Then there exists some $T>0$, such that (1.2) has a unique solution m in $C([0,T);B^{\frac{1}{2}}_{2,1})\cap C^1([0,T);B^{-\frac{1}{2}}_{2,1}).$

Proof

In order to prove Theorem 3.1, we proceed as the following five steps.

Step 1 First, we construct approximate solutions which are smooth solutions of some linear equations. Starting for $m_0(t,x)\triangleq m(0,x)=m_0$, we define by induction sequences $(m_{n})_{n\in {\mathbb {N}}}$ by solving the following linear transport equations:

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t}m_{n+1}-\partial _{x}u_{n}\partial _{x}m_{n+1} &{}=\frac{1}{2}(\partial _{x}u_n)^2-\frac{1}{2}(u_n-m_{n})^2 \\ &{}=u_{n}m_{n} +\frac{1}{2}(\partial _{x}u_n)^2-\frac{1}{2}u_n^2-\frac{1}{2}m_{n}^2 \\ &{}=F(m_{n},u_{n}),\\ m_{n+1}(t,x)|_{t=0}=S_{n+1}m_{0}.\\ \end{array} \right. \end{aligned}$$

(3.1)

We assume that $m_n\in L^{\infty }(0,T;B^{\frac{1}{2}}_{2,1})$. Note that $B^{\frac{1}{2}}_{2,1}$ is an algebra and $B^{\frac{1}{2}}_{2,1}\hookrightarrow L^\infty$, which leads to $F(m_{n},u_{n})\in L^{\infty }(0,T;B^{\frac{1}{2}}_{2,1})$. Hence, from Lemma 2.11, the Eq. (3.1) has a global solution $m_{n+1}$ which belongs to $C([0,T);B^{\frac{1}{2}}_{2,1})\cap C^1([0,T);B^{-\frac{1}{2}}_{2,1})$ for all positive T.

Step 2 Next, we are going to find some positive T such that for this fixed T the approximate solutions are uniformly bounded on [0, T]. We define that $U_{n}(t)\triangleq \int ^{t}_{0}\Vert m_n(t')\Vert _{B^{\frac{1}{2}}_{2,1}}dt'$. By Lemma 2.10, we infer that

$$\begin{aligned} \Vert m_{n+1}\Vert _{B^{\frac{1}{2}}_{2,1}}&\le e^{C\int _{0}^{t}\Vert \partial ^2_{x}u_n\Vert _{B^{\frac{1}{2}}_{2,1}}dt'} \bigg (\Vert S_{n+1}m_0\Vert _{B^{\frac{1}{2}}_{2,1}}\nonumber \\&\quad +\int ^t_0 e^{-C\int _{0}^{t'}\Vert \partial ^2_{x}u_n\Vert _{B^{\frac{1}{2}}_{2,1}}d\tau } \Vert F(m_{n},u_{n})\Vert _{B^{\frac{1}{2}}_{2,1}}dt'\bigg ) \nonumber \\&\le e^{CU_{n}(t)}\bigg (\Vert S_{n+1}m_0\Vert _{B^{\frac{1}{2}}_{2,1}} +\int ^t_0e^{-CU_{n}(t')}\Vert F(m_{n},u_{n})\Vert _{B^{\frac{1}{2}}_{2,1}}dt'\bigg ). \end{aligned}$$

(3.2)

Since $B^{\frac{1}{2}}_{2,1}$ is an algebra and $B^{\frac{1}{2}}_{2,1}\hookrightarrow L^{\infty }$, we deduce that

$$\begin{aligned}&\left\| \frac{1}{2}(\partial _{x}u_n)^2-\frac{1}{2}(u_n-m_{n})^2\right\| _{B^{\frac{1}{2}}_{2,1}} \nonumber \\&\quad \le \frac{1}{2}\Vert (\partial _{x}u_n)^2\Vert _{B^{\frac{1}{2}}_{2,1}} +\frac{1}{2}\Vert (u_n-m_{n})^2\Vert _{B^{\frac{1}{2}}_{2,1}} \nonumber \\&\quad \le \Vert \partial _{x}u_n\Vert _{B^{\frac{1}{2}}_{2,1}}\Vert \partial _{x}u_n\Vert _{L^{\infty }} +\Vert u_n-m_{n}\Vert _{B^{\frac{1}{2}}_{2,1}}\Vert u_n-m_{n}\Vert _{L^{\infty }} \nonumber \\&\quad \le C\Vert m_{n}\Vert ^2_{B^{\frac{1}{2}}_{2,1}}. \end{aligned}$$

(3.3)

Plugging (3.3) into (3.2), we obtain

$$\begin{aligned} \Vert m_{n+1}\Vert _{B^{\frac{1}{2}}_{2,1}}&\le e^{CU_{n}(t)}\bigg (\Vert S_{n+1}m_0\Vert _{B^{\frac{1}{2}}_{2,1}} +C\int ^t_0e^{-CU_{n}(t')}\Vert m_n\Vert ^2_{B^{\frac{1}{2}}_{2,1}}dt'\bigg ) \nonumber \\&\le e^{CU_{n}(t)}\bigg (C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}} +C\int ^t_0e^{-CU_{n}(t')}\Vert m_n\Vert ^2_{B^{\frac{1}{2}}_{2,1}}dt'\bigg ), \end{aligned}$$

(3.4)

where we take $C\ge 1.$

We fix a $T>0$ such that $2C^2T\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}<1.$ Suppose that

$$\begin{aligned} \Vert m_{n}(t)\Vert _{B^{\frac{1}{2}}_{2,1}}\le \frac{C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}{1-2C^2\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}t} \le \frac{C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}{1-2C^2\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}T}\triangleq {\mathbf {M}},~~~~\forall t\in [0,T]. \end{aligned}$$

(3.5)

Since $U_{n}(t)=\int ^{t}_{0}\Vert m_n(\tau )\Vert _{B^{\frac{1}{2}}_{2,1}}d\tau$, it follows that

$$\begin{aligned} e^{CU_{n}(t)-CU_{n}(t')}\le&\exp \left\{ \int ^{t}_{t'}\frac{C^2\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}{1-2C^2\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}t} d\tau \right\} \nonumber \\ \le&\exp \left\{ -\frac{1}{2}\int ^{t}_{t'} \frac{d\left( 1-2C^2\tau \Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}\right) }{1-2C^2\tau \Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}} \right\} \nonumber \\ =&\left( \frac{1-2C^2t'\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}{1-2C^2t\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}\right) ^{\frac{1}{2}}. \end{aligned}$$

(3.6)

Set $U_{n}(t')=0$ when $t'=0$. We obtain

$$\begin{aligned} e^{CU_{n}(t)} =&\left( \frac{1}{1-2C^2t\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}\right) ^{\frac{1}{2}}. \end{aligned}$$

(3.7)

By using (3.5), (3.6) and (3.7), we have

$$\begin{aligned}&\Vert m_{n+1}(t)\Vert _{B^{\frac{1}{2}}_{2,1}} \nonumber \\&\quad \le Ce^{CU_{n}(t)}\Vert m_0\Vert _{{B^{\frac{1}{2}}_{2,1}}} +C\int ^t_0e^{CU_{n}(t)-CU_{n}(t')}\Vert m_n(t')\Vert _{B^{\frac{1}{2}}_{2,1}}^{2}dt' \nonumber \\&\quad \le \left( \frac{1}{1-2C^2t\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}^{2}}\right) ^{\frac{1}{2}} \left\{ C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}+\int ^t_0\left( \frac{C^3\Vert m_0\Vert ^{2}_{B^{\frac{1}{2}}_{2,1}}}{\left( 1-2C^2t'\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}\right) ^{1+\frac{1}{2}}}\right) dt'\right\} \nonumber \\&\quad \le \left( \frac{1}{1-2C^2t\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}\right) ^{\frac{1}{2}} \left\{ C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}-\frac{C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}{2} \int ^t_0\frac{d\left( 1-2C^2t'\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}\right) }{\left( 1-2C^2t'\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}\right) ^{1+\frac{1}{2}}}\right\} \nonumber \\&\quad \le \left( \frac{1}{1-2C^2t\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}\right) ^{\frac{1}{2}} \left\{ C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}+C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}} \left( \frac{1}{1-2C^2t'\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}\right) ^{\frac{1}{2}}|^t_0\right\} \nonumber \\&\quad = \frac{C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}{1-2C^2t\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}} \nonumber \\&\quad \le \frac{C\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}{1-2C^2T\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}}={\mathbf {M}}. \end{aligned}$$

(3.8)

Thus, $(m_{n})_{n \in {\mathbb {N}}}$ is uniformly bounded in $L^{\infty }(0,T; B^{\frac{1}{2}}_{2,1})$.

Step 3 From now on, we are going to prove that $m_n$ is a Cauchy sequence in $L^{\infty }(0,T;B^{-\frac{1}{2}}_{2,\infty })$. For this purpose, we deduce from (3.1) that

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t}(m_{n+l+1}-m_{n+1})-\partial _{x}u_{n+l}\partial _{x}(m_{n+l+1}-m_{n+1})\\ \quad =\partial _{x}(u_{n+l}-u_{n})\partial _{x}m_{n+l} +\frac{1}{2}\partial _{x}(u_{n+l}-u_{n})\partial _{x}(u_{n+l}+u_{n}) \\ \qquad -\, \frac{1}{2}(u_{n+l}-u_{n}-m_{n+l}+m_{n})(u_{n+l}+u_{n}-m_{n+l}-m_{n}) \\ \quad =\partial _{x}(u_{n+l}-u_{n})\partial _{x}[m_{n+l}+\frac{1}{2}(u_{n+l}+u_{n})] -\frac{1}{2}(u_{n+l}-u_{n})(u_{n+l}+u_{n}-m_{n+l}-m_{n}) \\ \qquad +\,\frac{1}{2}(m_{n+l}-m_{n})(u_{n+l}+u_{n}-m_{n+l}-m_{n}) \\ \quad =\partial _{x}(u_{n+l}-u_{n})\partial _{x}R_{n,l}^{1} -\frac{1}{2}(u_{n+l}-u_{n})R_{n,l}^{2} +\frac{1}{2}(m_{n+l}-m_{n})R_{n,l}^{2},\\ m_{n+l+1}-m_{n+1}|_{t=0}=(S_{n+l+1}-S_{n+1})m_{0},\\ \end{array} \right. \end{aligned}$$

(3.9)

where

$$\begin{aligned}&R_{n,l}^{1}=m_{n+l}+\frac{1}{2}(u_{n+l}+u_{n}),\\&R_{n,l}^{2}=u_{n+l}+u_{n}-m_{n+l}-m_{n}. \end{aligned}$$

By Lemma 2.10 and using the fact that $m_n$ is bounded in $L^{\infty }(0,T;B^{\frac{1}{2}}_{2,1})$, we infer that

$$\begin{aligned}&\Vert m_{n+l+1}(t)-m_{n+1}(t)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\nonumber \\&\quad \le C_{T}\bigg (\Vert (S_{n+l+1}-S_{n+1})m_0\Vert _{B^{-\frac{1}{2}}_{2,\infty }} +\int ^t_0 \Vert \partial _{x}(u_{n+l}-u_{n})\partial _{x}R_{n,l}^{1}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} \nonumber \\&\qquad +\Vert \frac{1}{2}(u_{n+l}-u_{n})R_{n,l}^{2}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} +\Vert \frac{1}{2}(m_{n+l}-m_{n})R_{n,l}^{2}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} dt'\bigg ). \end{aligned}$$

(3.10)

Taking advantage of Lemma 2.4, we have

$$\begin{aligned}&\Vert \partial _{x}(u_{n+l}-u_{n})\partial _{x}R_{n,l}^{1}\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\nonumber \\&\quad \le \Vert \partial _{x}(u_{n+l}-u_{n})\Vert _{B^{\frac{1}{2}}_{2,1}} \Vert \partial _{x}\left[ m_{n+l}+\frac{1}{2}(u_{n+l}+u_{n})\right] \Vert _{B^{-\frac{1}{2}}_{2,\infty }} \nonumber \\&\quad \le \Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,1}}\Vert m_{n+l} +\frac{1}{2}(u_{n+l}+u_{n})\Vert _{B^{\frac{1}{2}}_{2,\infty }} \nonumber \\&\quad \le C\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,1}}\left( \Vert m_{n+1}\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert u_{n+l}\Vert _{B^{\frac{1}{2}}_{2,1}}+\Vert u_{n}\Vert _{B^{\frac{1}{2}}_{2,1}}\right) \nonumber \\&\quad \le 3C{\mathbf {M}}\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,1}}, \end{aligned}$$

(3.11)

$$\begin{aligned}&\left\| \frac{1}{2}(u_{n+l}-u_{n})R_{n,l}^{2}\right\| _{B^{-\frac{1}{2}}_{2,\infty }}\nonumber \\&\quad \le \Vert u_{n+l}-u_{n}\Vert _{B^{ \frac{1}{2}}_{2,1}}\Vert u_{n+l}+u_{n}-m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} \nonumber \\&\quad \le C\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,1}}\left( \Vert m_{n+l}\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert m_{n}\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert u_{n+l}\Vert _{B^{\frac{1}{2}}_{2,1}}+\Vert u_{n}\Vert _{B^{\frac{1}{2}}_{2,1}}\right) \nonumber \\&\quad \le 4C{\mathbf {M}}\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,1}}, \end{aligned}$$

(3.12)

$$\begin{aligned}&\left\| \frac{1}{2}(m_{n+l}-m_{n})R_{n,l}^{2}\right\| _{B^{-\frac{1}{2}}_{2,\infty }}\nonumber \\&\quad \le \Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2} }_{2,1}}\Vert u_{n+l}+u_{n}-m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} \nonumber \\&\quad \le C\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,1}}\left( \Vert m_{n+l}\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert m_{n}\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert u_{n+l}\Vert _{B^{\frac{1}{2}}_{2,1}}+\Vert u_{n}\Vert _{B^{\frac{1}{2}}_{2,1}}\right) \nonumber \\&\quad \le 4C{\mathbf {M}}\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,1}}, \end{aligned}$$

(3.13)

where $C>1.$ Plugging (3.11)–(3.13) into (3.10) yields that

$$\begin{aligned} \Vert m_{n+l+1}(t)-m_{n+1}(t)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}&\le C_{T}\bigg (\Vert (S_{n+l+1}-S_{n+1})m_0\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\nonumber \\&\quad +\int ^t_0 11C{\mathbf {M}}\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,1}}dt'\bigg ). \end{aligned}$$

(3.14)

Applying Lemma 2.6 to the above inequality, we have

$$\begin{aligned}&\Vert m_{n+l+1}(t)-m_{n+1}(t)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\le C_{T}\left( \Vert (S_{n+l+1}-S_{n+1})m_0\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\right. \nonumber \\&\qquad +\int ^t_0 11C{\mathbf {M}} \Vert m_{n+l}(t')-m_{n}(t')\Vert _{B^{-\frac{1}{2}}_{2,\infty }} \ln \left( e+\frac{\Vert m_{n+l}(t')-m_{n}(t')\Vert _{B^{\frac{1}{2}}_{2,1}}}{\Vert m_{n+l}(t')-m_{n}(t')\Vert _{B^{-\frac{1}{2}}_{2,\infty }}}\right) dt' \left. \right) . \end{aligned}$$

(3.15)

Since

$$\begin{aligned} \left\| \sum _{q=n+1}^{n+l}\Delta _{q}m_{0}\right\| _{B^{-\frac{1}{2}}_{2,\infty }}\le C2^{-n}\Vert m_{0}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} , \end{aligned}$$

and that $(m_n)_{n\in {\mathbb {N}}}$ is uniformly bounded in $L^{\infty }([0,T];B^{s}_{p,r})$, then it follows that

$$\begin{aligned}&\Vert m_{n+l+1}(t)-m_{n+1}(t)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\\&\quad \le \widetilde{C_T}\left( 2^{-n}+\int ^{t}_{0}\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} \ln \left( e+\frac{C}{\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,\infty }}}\right) d\tau \right) \\&\quad \le \widetilde{C_T}\left( 2^{-n}+\int ^{t}_{0}\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} \left( 1-\ln \frac{\Vert m_{n+l}-m_{n}\Vert _{B^{-\frac{1}{2}}_{2,\infty }}}{C}\right) d\tau \right) . \end{aligned}$$

Noticing that the function $x(1-\ln {\frac{x}{C}})$ is nondecreasing in $x\in [0,C)$, we get

$$\begin{aligned}&\Vert m_{n+l+1}(t)-m_{n+1}(t)\Vert _{L_{t}^{\infty }(B^{-\frac{1}{2}}_{2,\infty })} \nonumber \\&\quad \le \widetilde{C_T}\left( 2^{-n}+\int ^{t}_{0}\Vert m_{n+l}-m_{n}\Vert _{L_{t}^{\infty }(B^{-\frac{1}{2}}_{2,\infty })} \left( 1-\ln \frac{\Vert m_{n+l}-m_{n}\Vert _{L_{t}^{\infty }(B^{-\frac{1}{2}}_{2,\infty })}}{C}\right) d\tau \right) . \end{aligned}$$

(3.16)

Let $g_n(t)=\sup _{l\in {\mathbb {N}}}\Vert m_{n+l}(t)-m_{n}(t)\Vert _{L_{t}^{\infty }(B^{-\frac{1}{2}}_{2,\infty })}.$ Noticing that the function $x(1-\ln {\frac{x}{C}})$ is nondecreasing in $x\in [0,C)$, we get

$$\begin{aligned} g_{n+1}(t)\le \widetilde{C_{T}}\bigg (2^{-n}+\int ^T_0g_n(t)\left( 1-\ln \frac{g_n(t)}{C}\right) dt\bigg ), \end{aligned}$$

(3.17)

which along with the Fatou–Lebesgue theorem leads to

$$\begin{aligned} {\widetilde{g}}(t)\triangleq \limsup _{n\rightarrow \infty }g_{n+1}(t)\le C\int ^T_0{\widetilde{g}}(t)\left( 1-\ln \frac{{\widetilde{g}}(t)}{C}\right) dt. \end{aligned}$$

(3.18)

By Lemma 2.8, we infer that $m_{n+l+1}(t)-m_{n+1}(t)=0$ as $n\rightarrow \infty$. In other words, $(m_n)_{n\in {\mathbb {N}}}$ is a Cauchy sequence in $L^{\infty }(0,T;B^{-\frac{1}{2}}_{2,\infty })$ and converges to some limit function $m\in L^{\infty }(0,T;B^{-\frac{1}{2}}_{2,\infty })$.

Step 4 We now prove the existence of solutions. We prove that m belongs to $C([0,T);B^{\frac{1}{2}}_{2,1})\cap C^1([0,T);B^{-\frac{1}{2}}_{2,1})$ and satisfies the Eq. (1.2) in the sense of distributions. Since $(m_n)_{n\in {\mathbb {N}}}$ is uniformly bounded in $L^\infty (0,T;B^{\frac{1}{2}}_{2,1})$, the Fatou property for the Besov spaces ensures that $m \in L^\infty (0,T;B^{\frac{1}{2}}_{2,1})$.

Taking limit in the Eq. (3.1), we conclude that m is indeed a solution of (1.2). Note that $m\in L^\infty (0,T;B^{\frac{1}{2}}_{2,1}).$ Then

$$\begin{aligned}&\left\| \frac{1}{2}(\partial _{x}u)^2-\frac{1}{2}(u-m)^2\right\| _{B^{\frac{1}{2}}_{2,1}}\nonumber \\&\quad \le \frac{1}{2}\Vert (\partial _{x}u)^2\Vert _{B^{\frac{1}{2}}_{2,1}} +\frac{1}{2}\Vert (u-m)^2\Vert _{B^{\frac{1}{2}}_{2,1}} \nonumber \\&\quad \le \Vert \partial _{x}u\Vert _{B^{\frac{1}{2}}_{2,1}}\Vert \partial _{x}u\Vert _{L^{\infty }} +\Vert u-m\Vert _{B^{\frac{1}{2}}_{2,1}}\Vert u-m\Vert _{L^{\infty }}\nonumber \\&\quad \le C\Vert m\Vert ^2_{B^{\frac{1}{2}}_{2,1}}. \end{aligned}$$

(3.19)

This means that the right-hand side of (1.2) also belongs to $L^\infty (0,T;B^{\frac{1}{2}}_{2,1}).$ Hence, according to Lemma 2.11, the function m belongs to $C([0,T);B^{\frac{1}{2}}_{2,1})$. Lemma 2.5 implies that $(4u-2u_x)m_x$ is bounded in $L^\infty (0,T;B^{-\frac{1}{2}}_{2,1}).$ Again using the equation (1.2) and high regularity of u, we see that $\partial _{t}u$ is in $C([0,T);B^{-\frac{1}{2}}_{2,1})$. Then, we know that $u\in C([0,T);B^{\frac{1}{2}}_{2,1})\cap C^1([0,T);B^{-\frac{1}{2}}_{2,1}).$

Step 5 Finally, we prove the uniqueness of strong solutions to (1.2). Suppose that $M=(1-\partial _x^2)u,~N=(1-\partial _x^2)v \in E^{s}_{p,r}$ are two solutions of (1.2). Set $W=M-N.$ Hence, we obtain that

$$\begin{aligned} \left\{ \begin{array}{llll} &{}\partial _{t}W-\partial _{x}u\partial _{x}W \\ &{}=\partial _{x}(u-v)\partial _{x}G^{1} -\frac{1}{2}(u-v)G^{2}+\frac{1}{2}W G^{2},\\ &{}W|_{t=0}=M(0,x)-N(0,x)=W(0),\\ \end{array} \right. \end{aligned}$$

(3.20)

where

$$\begin{aligned}&G^{1}=N+\frac{1}{2}(u+v),\\&G^{2}=u+v-M-N. \end{aligned}$$

We define that $U(t)\triangleq \int ^{t}_{0}\Vert m(t')\Vert _{B^{\frac{1}{2}}_{2,1}}dt'$. By Lemma 2.10 and using the fact that m is bounded in $L^{\infty }(0,T;B^{\frac{1}{2}}_{2,1})$, we infer that

$$\begin{aligned} \Vert W\Vert _{B^{-\frac{1}{2}}_{2,\infty }}&\le Ce^{CU(t)}\bigg (\Vert W(0)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}+\int ^t_0 e^{-CU(t')}\left( \left\| \frac{1}{2}\partial _{x}(u-v)\partial _{x}G^{1}\right\| _{B^{-\frac{1}{2}}_{2,\infty }}\right. \nonumber \\&\quad \left. \left. +\frac{1}{2}\Vert (u-v)G^{2}\Vert _{B^{-\frac{1}{2}}_{2,\infty }} +\frac{1}{2}\Vert WG^{2}\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\right) dt' \right) . \end{aligned}$$

(3.21)

Taking advantage of Lemma 2.4, we have

$$\begin{aligned} \left\| \frac{1}{2}\partial _{x}(u-v)\partial _{x}G^{1}\right\| _{B^{-\frac{1}{2}}_{2,\infty }}&\le \Vert \partial _{x}(u-v)\Vert _{B^{\frac{1}{2}}_{2,1}} \left\| \partial _{x}\left( N+\frac{1}{2}(u+v)\right) \right\| _{B^{-\frac{1}{2}}_{2,\infty }} \nonumber \\&\le \Vert W\Vert _{B^{-\frac{1}{2}}_{2,1}}\left\| N+\frac{1}{2}(u+v)\right\| _{B^{\frac{1}{2}}_{2,\infty }} \nonumber \\&\le C\Vert W\Vert _{B^{-\frac{1}{2}}_{2,1}}\left( \Vert N\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert u\Vert _{B^{\frac{1}{2}}_{2,1}}+\Vert v\Vert _{B^{\frac{1}{2}}_{2,1}}\right) \nonumber \\&\le 3C{\mathbf {M}}\Vert W\Vert _{B^{-\frac{1}{2}}_{2,1}}, \end{aligned}$$

(3.22)

$$\begin{aligned} \Vert (u-v)G^{2}\Vert _{B^{-\frac{1}{2}}_{2,\infty }}&\le \Vert u-v\Vert _{B^{\frac{1}{2}}_{2,1}}\Vert u+v-M-N\Vert _{B^{-\frac{1}{2}}_{2,\infty }} \nonumber \\&\le C\Vert W\Vert _{B^{-\frac{1}{2}}_{2,1}}\left( \Vert M\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert N\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert u\Vert _{B^{\frac{1}{2}}_{2,1}}+\Vert v\Vert _{B^{\frac{1}{2}}_{2,1}}\right) \nonumber \\&\le 4C{\mathbf {M}}\Vert W\Vert _{B^{-\frac{1}{2}}_{2,1}}, \end{aligned}$$

(3.23)

$$\begin{aligned} \Vert WG^{2}\Vert _{B^{-\frac{1}{2}}_{2,\infty }}&\le \Vert W\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\Vert u+v-M-N\Vert _{B^{\frac{1}{2}}_{2,1}} \nonumber \\&\le C\Vert W\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\left( \Vert M\Vert _{B^{\frac{1}{2}}_{2,1}}+\Vert N\Vert _{B^{\frac{1}{2}}_{2,1}} +\Vert u\Vert _{B^{\frac{1}{2}}_{2,1}}+\Vert v\Vert _{B^{\frac{1}{2}}_{2,1}}\right) \nonumber \\&\le 4C {\mathbf {M}}\Vert W\Vert _{B^{-\frac{1}{2}}_{2,1}}. \end{aligned}$$

(3.24)

Plugging (3.22)–(3.24) into (3.21) yields that

$$\begin{aligned}&e^{-CU(t)}\Vert W(t)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\nonumber \\&\quad \le C\Vert W(0)\Vert _{B^{-\frac{1}{2}}_{2,\infty }} +\int ^t_0 11C{\mathbf {M}}e^{-CU(t')}\Vert W(t')\Vert _{B^{-\frac{1}{2}}_{2,1}}dt'\nonumber \\&\quad \le C\Vert W(0)\Vert _{B^{-\frac{1}{2}}_{2,\infty }} +\int ^t_0 11C{\mathbf {M}}e^{-CU(t')}\Vert W(t')\Vert _{B^{-\frac{1}{2}}_{2,\infty }} \ln \left( e+\frac{\Vert W(t)\Vert _{B^{\frac{1}{2}}_{2,1}}}{\Vert |W(t)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}}\right) dt'. \end{aligned}$$

(3.25)

Now define ${\widetilde{W}}(t)\triangleq e^{-CU(t)}\Vert W(t)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}$. Since the function $xln(e+\frac{C}{x})$ is nondecreasing and m is bounded in $L^{\infty }(0,T;B^{\frac{1}{2}}_{2,1})$, it follows that

$$\begin{aligned} {\widetilde{W}}(t)&\le C_1\bigg ({\widetilde{W}}(0) +\int ^t_0{\widetilde{W}}(t')\ln \left( e+\frac{C_1}{{\widetilde{W}}(t')}\right) dt'\bigg )\nonumber \\&\le C_1\bigg ({\widetilde{W}}(0) +\int ^t_0{\widetilde{W}}(t')\left( 1-\ln \frac{{\widetilde{W}}(t')}{C_1}\right) dt'\bigg ). \end{aligned}$$

(3.26)

By virtue of Lemma 2.8 and Remark 2.9 with $\rho =\frac{{\widetilde{W}}(t)}{C_1}$, we verifies that

$$\begin{aligned} {\widetilde{W}}(t)\le C_1{\widetilde{W}}(0)^{\exp \{-C_1t\}} \end{aligned}$$

which leads to

$$\begin{aligned} \Vert W(t)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}\le C_2\Vert W(0)\Vert _{B^{-\frac{1}{2}}_{2,\infty }}{\exp \{C_2t\}} \le C_{2}\Vert W(0)\Vert _{B^{-\frac{1}{2}}_{2,1}}{\exp \{C_2T\}}. \end{aligned}$$

(3.27)

Taking advantage of the interpolation argument ensures that

$$\begin{aligned} \Vert W(t)\Vert _{L^{\infty }([0,T],B^{s'}_{2,1})}\le C_3\Vert W(0)\Vert _{B^{\frac{1}{2}}_{2,1}}^{\theta }{\exp \{C_2T\}}, \end{aligned}$$

(3.28)

where $\theta =\frac{1}{2}-s'\in (0,1]$. The above inequality implies the uniqueness.

This completes the proof of Theorem 3.1. $\square$

Next, we prove the solution of (1.2) guaranteed by Theorem 3.1 depends continuously on the initial data.

Theorem 3.2

Denote $\overline{{\mathbb {N}}}={\mathbb {N}}\cup {\infty }$. Let $(m_n)_{n\in \overline{{\mathbb {N}}}}$ be the corresponding solution of (1.2) guaranteed by Theorem 3.1 with the initial data $m^n_0(x)\in B^{\frac{1}{2}}_{2,1}$. If $m^n_ 0$ tends to $m^{\infty }_0$ in $B^{\frac{1}{2}}_{2,1}$, then $m^n(t,x)$ tends to $m^\infty (t,x) \in C([0,T];B^{\frac{1}{2}}_{2,1})$ with $2C^2T\Vert m_0\Vert _{B^{\frac{1}{2}}_{2,1}}<1.$

Proof

By Theorem 3.1, we can find $M > 0$ such that for all $n \in \overline{{\mathbb {N}}}$,

$$\begin{aligned}\sup _{n\in \overline{{\mathbb {N}}}}\Vert u^n\Vert _{L^{\infty } ([0,T];B^{\frac{1}{2}}_{2,1})} \le M.\end{aligned}$$

From (3.27), we have $\Vert m^n-m^\infty \Vert _{L^\infty (0,T;B^{-\frac{1}{2}}_{2,\infty } )}$ tends to zero as $n \rightarrow \infty$.

For fixed $\varphi \in B^{-\frac{1}{2}}_{2,1}$, we write

$$\begin{aligned} \langle m^n-m^\infty ,\varphi \rangle&=\langle S_{j}[m^n-m^\infty ],\varphi \rangle +\langle (Id-S_{j})[m^n-m^\infty ],\varphi \rangle \nonumber \\&=\langle m^n-m^\infty ,S_{j}\varphi \rangle +\langle m^n-m^\infty ,(Id-S_{j})\varphi \rangle . \end{aligned}$$

(3.29)

Direct computations show that

$$\begin{aligned} |\langle m^n-m^\infty ,S_{j}\varphi \rangle |\le CM\Vert m^n-m^\infty \Vert _{L^\infty ([0,T);B^{-\frac{1}{2}}_{2,\infty })} \Vert S_{j}\varphi \Vert _{B^{\frac{1}{2}}_{2,1}}, \end{aligned}$$

(3.30)

and

$$\begin{aligned} |\langle m^n-m^\infty ,(Id-S_{j})\varphi \rangle |\le CM\Vert \varphi -S_{j}\varphi \Vert _{B^{-\frac{1}{2}}_{2,1}}. \end{aligned}$$

(3.31)

Note that $\Vert \varphi -S_{j}\varphi \Vert _{B^{-\frac{1}{2}}_{2,1}}$ tends to zero as $j\rightarrow \infty$ and $\Vert m^n-m^\infty \Vert _{L^\infty ([0,T);B^{-\frac{1}{2}}_{2,1})}$ tends to zero as $n\rightarrow \infty$. Then (3.31) may be made arbitrarily small for j large enough. For fixed j, we then let n tend to infinity so that (3.30) tends to zero. Thus, we conclude that $\langle m^n-m^\infty ,\varphi \rangle$ tends to zero. Then we obtain $m^n$ tends to $m^\infty$ in ${\mathbb {C}}_{w}([0,T);B^{\frac{1}{2}}_{2,\infty }).$ Because $B^{\frac{1}{2}}_{2,\infty }\hookrightarrow B^{-\frac{1}{2}}_{2,1},$ we obtain $m^n$ tends to $m^\infty$ in ${\mathbb {C}}([0,T);B^{-\frac{1}{2}}_{2,1}).$

Note that for all $m=u-u_{xx},{\widetilde{m}}={\widetilde{u}}-{\widetilde{u}}_{xx},\in B^{\frac{1}{2}}_{2,1}$,

$$\begin{aligned}&\Vert u_{x}-{\widetilde{u}}_{x}\Vert _{B^{\frac{1}{2}}_{2,1}} \le C\Vert m-{\widetilde{m}}\Vert _{B^{-\frac{1}{2}}_{2,1}}, \end{aligned}$$

(3.32)

$$\begin{aligned}&\left\| \frac{1}{2}u^2_{x}-\frac{1}{2}(u-m)^2 -\frac{1}{2}{\widetilde{u}}^2_{x}+\frac{1}{2} ({\widetilde{u}}-{\widetilde{m}})^2\right\| _{B^{\frac{1}{2}}_{2,1}}\nonumber \\&\quad \le \frac{1}{2}\Vert u^2_{x}-{\widetilde{u}}^2_{x}\Vert _{B^{\frac{1}{2}}_{2,1}}+ \frac{1}{2}\Vert (u-m)^2-({\widetilde{u}}-{\widetilde{m}})^2\Vert _{B^{\frac{1}{2}}_{2,1}} \nonumber \\&\quad \le \frac{1}{2}\Vert (u_{x}-{\widetilde{u}}_{x})(u_{x}+{\widetilde{u}}_{x})\Vert _{B^{\frac{1}{2}}_{2,1}}+ \frac{1}{2}\Vert (u-{\widetilde{u}}-m+{\widetilde{m}})(u+{\widetilde{u}}-m-{\widetilde{m}})\Vert _{B^{\frac{1}{2}}_{2,1}} \nonumber \\&\quad \le CM\Vert m-{\widetilde{m}}\Vert _{B^{-\frac{1}{2}}_{2,1}}. \end{aligned}$$

(3.33)

Then by Lemma 2.12 and (3.19), we have that $u^n$ tends to $u \in C([0, T]; B^{\frac{1}{2}}_{2,1})$. $\square$

4 A blow-up criterion

After establishing local well-posedness theory, a natural question is whether the corresponding solution exists globally in time or not. This section is devoted to investigating a blow-up criterion for (1.2). At first, we show a conservation law and an a priori estimate for strong solutions to (1.2).

Lemma 4.1

Let $u_0\in H^s,~s> \frac{5}{2}.$ Then the corresponding solution u to (1.2) has constant energy integral

$$\begin{aligned}\int _{{\mathbb {R}}}(u_{x}^2+u_{xx}^2)dx =\int _{{\mathbb {R}}}\left( (u_0')^2+(u_0'')^2\right) dx=\Vert u'_0\Vert ^2_{H^1}.\end{aligned}$$

Proof

Arguing by density, it suffices to consider the case where $u\in C^{\infty }_0.$ Applying integration by parts, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}}u_{x}m_{x} dx=\int _{{\mathbb {R}}}u_{x}(u_{x}- u_{xxx}) dx=\int _{{\mathbb {R}}}u_{x}^2dx+\int _{{\mathbb {R}}} u_{xx}^2dx. \end{aligned}$$

Taking advantage of Lemmas 4.1 and 4.2, we infer that

$$\begin{aligned}&\frac{d}{dt}\int _{{\mathbb {R}}}u_{x}m_{x} dx\\&\quad =\int _{{\mathbb {R}}}(\partial _tu _{x} m_{x}+\partial _tm_{x}u_{x})dx \\&\quad =2\int _{{\mathbb {R}}}\partial _tm_{x} u_{x}dx =2\int _{{\mathbb {R}}}u_{x}[u_{x}m_{xx}+2u_{xx}m_{x}]dx \\&\quad =-2\int _{{\mathbb {R}}}u^2_{x}m_{xx}+2u_{x}u_{xx}m_{x}dx \\&\quad =0. \end{aligned}$$

$\square$

Lemma 4.2

Let $u_0\in H^s,~s>\frac{5}{2}$, and let T be the maximal existence time of the corresponding solution u to (1.2). Then we have

$$\begin{aligned} \Vert u\Vert _{L^{2}}\le \Vert u_0\Vert _{L^{2}}+2\Vert u'_0\Vert ^2_{H^1}T. \end{aligned}$$

(4.1)

Proof

Arguing by density, it suffices to consider the case where $u\in C^{\infty }_0.$

Note that $G(x)=\frac{1}{2}e^{-|x|}$ and $G(x)\star f =(1-\partial _x^2)^{-1}f$ for all $f \in L^2({\mathbb {R}})$ and $G \star m=u$. Then we can rewrite (1.1) as follows:

$$\begin{aligned} u_t-\frac{1}{2}u^2_x=G*\left[ u_x^2+\frac{1}{2} u_{xx}^{2}\right] . \end{aligned}$$

(4.2)

By (4.2), we infer that

$$\begin{aligned}&\frac{d}{dt}\int _{{\mathbb {R}}}u^2dx\\&\quad =2\int _{{\mathbb {R}}}\partial _tu udx \\&\quad =2\int _{{\mathbb {R}}}u\left[ \frac{1}{2}u^2_{x}+G*\left( u_{x}^2+\frac{1}{2}u_{xx}^2\right) \right] dx \\&\quad \le \Vert u\Vert _{L^2}\left[ \Vert u_{x}\Vert _{L^{2}}\Vert u_{x}\Vert _{L^{\infty }}+\Vert G\Vert _{L^{2}} (2\Vert u_{x}\Vert ^2_{L^{2}}+\Vert u_{xx}\Vert ^2_{L^{2}})\right] \\&\quad \le 4\Vert u\Vert _{L^2}\Vert u'_{0}\Vert ^2_{H^{1}}. \end{aligned}$$

Thus we have

$$\begin{aligned} \Vert u\Vert _{L^{2}}\le \Vert u_0\Vert _{L^{2}}+2\Vert u'_0\Vert ^2_{H^1}t\le \Vert u_0\Vert _{H^{2}}+2\Vert u'_0\Vert ^2_{H^1}T. \end{aligned}$$

(4.3)

$\square$

Remark 4.3

From Lemmas 4.1 and 4.2, we obtain

$$\begin{aligned} \Vert u\Vert _{H^2}\le \Vert u\Vert _{L^2}+\Vert u_{x}\Vert _{L^2}+\Vert u_{xx}\Vert _{L^2}\le \Vert u_0\Vert _{H^{2}}+2\Vert u'_0\Vert ^2_{H^1}T. \end{aligned}$$

(4.4)

Then we present a blow-up criterion for (1.2).

Lemma 4.4

Let $u_0(x)\in H^s,~ s> \frac{5}{2},$ and let T be the maximal existence time of the solution u(x, t) to (1.2) with the initial data $u_0(x)$. Then the corresponding solution blows up in finite time if and only if

$$\begin{aligned} \liminf _{t\rightarrow T}\inf _{x\in {\mathbb {R}}}m=-\infty . \end{aligned}$$

Proof

Arguing by density, it suffices to consider the case where $u\in C^{\infty }_0$.

A direct computation yields

$$\begin{aligned} \Vert m_{x}\Vert ^2_{L^2}&=\int _{{\mathbb {R}}}(u_{x}- u_{xxx})^2dx\\&=\int _{{\mathbb {R}}}(u_x^2+2u_{xx}^2+ u_{xxx}^2)dx. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert u_{x}\Vert _{H^2}\le \Vert m_{x}\Vert _{L^2}\le 2\Vert u_{x}\Vert _{H^2}. \end{aligned}$$

(4.5)

Differentiating both sides of (1.2) with respect to x, taking $L^2$ inner product with $m_x$, and then integrating by parts, we obtain

$$\begin{aligned}&\frac{d}{dt}\int _{{\mathbb {R}}} m_{x}^2dx\\&\quad = \int _{{\mathbb {R}}}2m_{x} \partial _{t}m_{x} dx\\&\quad = 2\int _{{\mathbb {R}}}m_{x}[2(u-m)m_{x}+u_{x}m_{xx}]dx\\&\quad = \int _{{\mathbb {R}}}[4(u-m)m^2_{x}+2u_{x}m_{x}m_{xx}]dx\\&\quad = \int _{{\mathbb {R}}}[4(u-m)m^2_{x}-u_{xx}m^2_{x}]dx\\&\quad =\int _{{\mathbb {R}}}3(u-m)m^2_{x}dx. \end{aligned}$$

Suppose that m is bounded from above on [0, T) and $T<\infty$. By (4.5) and Lemmas 4.1–4.2, we get

$$\begin{aligned}&\frac{d(\Vert m_{x}\Vert ^2_{L^2})}{dt}\nonumber \\&\quad =\int _{{\mathbb {R}}}[3(u-m)m^2_{x}]dx\nonumber \\&\quad \le C\Vert m_{x}\Vert _{L^2}^2+3\Vert u\Vert _{L^\infty }\Vert m_{x}\Vert ^2_{L^{2}}\nonumber \\&\quad \le C_1\Vert m_{x}\Vert ^2_{L^2}, \end{aligned}$$

(4.6)

here $C_1~>0.$ An application of Gronwall’s inequality yields

$$\begin{aligned} \Vert m\Vert _{H^1}\le e^{C_1t}\Vert m_0\Vert _{H^1},~~\forall ~t\in [0,T). \end{aligned}$$

(4.7)

So in view of (4.5) and (4.7), we obtain that if m is bounded from above on [0, T), then so does the $H^2$-norm of $u_{x}$, which contradicts the assumption that $T < \infty$ is the maximal existence time. This completes the proof. $\square$

5 Global existence

In this section, we present a global existence result for the Cauchy problem (1.2).

Theorem 5.1

Assume that $u_0\in H^4$ is such that the associated potential $m'_0=u'_{0}-u'''_{0}$ satisfies $m'_0(x_{0})>0$, $m'_0(x)\ge 0$ on $(-\infty ,x_{0})$ and $m'_0(x)\le 0$ on $(x_{0},+\infty )$ for some point $x_{0}\in {\mathbb {R}}.$ Then the corresponding solution to (1.2) exists globally in time.

Proof

Applying Theorem 3.1 and a simple density argument, it suffices to consider $u_{0}\in H^4$ to prove the above theorem. Given $u_0 \in H^4$, $T^*$ is the maximal existence time of the corresponding solution to (1.2) with the initial data $u_0$.

We consider the following initial value problem

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{dq(t,x)}{dt}=-u_{x}(t,q(t,x)),~~t\in [0,T^*),~~x\in {\mathbb {R}},\\ q(0,x)=x,~~x\in {\mathbb {R}}. \end{array} \right. \end{aligned}$$

(5.1)

By applying classical results in the theory of ordinary differential equations, we infer that (5.1) has a unique solution $q \in C^1([0,T)\times {\mathbb {R}};{\mathbb {R}})$. Moreover, the map $q(\cdot ,t)$ is an increasing diffeomorphism of ${\mathbb {R}}$ with

$$\begin{aligned} q_x(t,x)=\exp \left( \int ^t_0 -u_{xx}(\tau ,q(\tau ,x))d\tau \right) >0, ~~~\forall ~(t,x)\in [0,T)\times {\mathbb {R}}. \end{aligned}$$

Hence, from (1.2), the following identity can be proved:

$$\begin{aligned} m_{x}(t,q(t,x))q_x(t,x)=m'_0(x)e^{\int ^t_0 u_{xx}dt'},~~\forall ~(t,x)\in [0,T)\times {\mathbb {R}}. \end{aligned}$$

(5.2)

In fact, a direct computation yields

$$\begin{aligned}&\frac{d}{dt}\{m_{x}(t,q(t,x))q_x(t,x)\}\nonumber \\&\quad =m_{tx}q_x+m_{xx}q_tq_x-m_{x}q_{xt}\nonumber \\&\quad =q_x(m_{tx}-u_{x}m_{xx}-u_{xx}m_{x})\nonumber \\&\quad =q_xu_{xx}m_{x}. \end{aligned}$$

(5.3)

Applying Gronwall’s inequality, we obtain (5.2).

Since q(t, x) is an increasing diffeomorphism of ${\mathbb {R}}$ as long as $t\in [0,T)$, we deduce

$$\begin{aligned} \left\{ \begin{array}{ll} m'(t,x)\ge 0,~if~~~x\le q(t,x_{0}),\\ m'(t,x)\le 0,~if~~~x\ge q(t,x_{0}). \end{array} \right. \end{aligned}$$

(5.4)

Using the fact that the flow q(t, x) is a differmorphism and $m_0(x_0) > 0$, and by (5.4) we see that $x_0$ is the maximum value point. (5.4) tells us that m(t) will increase monotonously at the interval $(-\infty , q(t,x_0))$ along the flow and decrease monotonously at the interval $(q(t,x_0), +\infty )$ along the flow. Moreover, since $m(t,q(t,x_0))$ belongs to $H^s$, $m(t,q(t,x_0))$ will not be less than zero. Otherwise, it will contradict with the decay of infinity in $H^s$. As a result, Theorem 4.4 ensure that the solution u(t, x) exists globally in time. This completes the proof of the theorem. $\square$

Applying $D_{x}$ to both sides of the Eq. (4.2) yield

$$\begin{aligned} (u_{x})_t-u_{x}u_{xx}=\partial _{x}G*\left[ u_x^2+\frac{1}{2} u_{xx}^{2}\right] , \end{aligned}$$

(5.5)

which indicates that $u_{x}$ satisfies the CH Eq. (1.4). Thus solution to (4.2) or (5.5) are actually the velocity potentials of the solution to the CH equation. Then by comparing the global existence of these two equations, we obtain the following Remark.

Remark 5.2

The Eq. (1.2) and the CH equation are truly different for the global existence. As for the CH equation, the global strong solutions exist globally in time under some conditions that $m_{0}$ don’t change the sign and that $m_{0}$ satisfies $m_{0}\le 0$ on $(-\infty , x_{0})$ and $m_{0}\ge 0$ on $(x_{0},+\infty , )$ for some point $x_{0}\in {\mathbb {R}}$ [9]. But as for the equation (4.2), there only exist zero solutions under some conditions that $m'_{0}$ don’t change the sign. The corresponding solutions to (1.2) exist globally with some certain conditions that $m_{0}$ satisfies $m_{0}(x_{0})>0$, $m'_{0}\ge 0$ on $(-\infty , x_{0})$ and $m'_{0}\le 0$ on $(x_{0},+\infty , )$ for some point $x_{0}\in {\mathbb {R}}$.

6 Blow-up phenomena

In this section we prove that there are some initial data for which the corresponding solutions to (1.2) with some certain conditions will blow up in finite time.

Theorem 6.1

Assume that $u_0\in H^s, ~~s>\frac{5}{2}$. And $m'_{0}$ satisfy $m'_{0}\le 0$ on $(-\infty , x_{0})$ and $m'_{0}\ge 0$ on $(x_{0},+\infty , )$ for some point $x_{0}\in {\mathbb {R}},$ then the corresponding solution of (1.2) blows up in finite time.

Proof

Applying Theorem 3.1 and a simple density argument, it suffices to consider $u_{0}\in H^4$ to prove the above theorem. Given $u_0 \in H^4$, $T^*$ is the maximal existence time of the corresponding solution to (1.2) with the initial data $u_0$.

Since q(t, x) is an increasing diffeomorphism of ${\mathbb {R}}$ as long as $t\in [0,T)$, we deduce

$$\begin{aligned} \left\{ \begin{array}{ll} m'(t,x)\le 0, \quad if~~~x\le q(t,x_{0}),\\ m'(t,x)\ge 0, \quad if ~~~x\ge q(t,x_{0}). \end{array} \right. \end{aligned}$$

(6.1)

Because of $u_{x}=G\star m_{x}$ where $G>0$, we can write $u_{x}(t,x)$ and $u_{xx}(t,x)$ as

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

(6.2)

$$\begin{aligned} u_{xx}(t,x)&=-\frac{e^{-x}}{2} \int ^{x}_{-\infty }e^{\xi }m_{\xi }(t,\xi )d\xi + \frac{e^{x}}{2} \int ^{\infty }_{x}e^{-\xi }m_{\xi }(t,\xi )d\xi . \end{aligned}$$

(6.3)

Consequently

$$\begin{aligned} (u_{x}+u_{xx})(t,x)= & {} e^{x}\int ^{\infty }_{x}e^{-\xi }m_{\xi }(t,\xi )d\xi , \\ (u_{x}-u_{xx})(t,x)= & {} e^{-x}\int _{-\infty }^{x}e^{-\xi }m_{\xi }(t,\xi )d\xi , \end{aligned}$$

for all $t\ge 0.$

Differentiating (4.2) with respect to x by two times, we find

$$\begin{aligned} u_{txx}-u_{x}u_{xxx}=\frac{1}{2}u^2_{xx}-u^2_{x}+G*\left[ u_x^2+\frac{1}{2} u_{xx}^{2}\right] . \end{aligned}$$

(6.4)

Combining the inequality $G*[u_x^2+\frac{1}{2} u_{xx}^{2}]\ge \frac{1}{2} u_{x}^{2}$ with (6.4), we deduce that

$$\begin{aligned} u_{txx}-u_{x}u_{xxx}\ge \frac{1}{2}u^2_{xx}-\frac{1}{2}u^2_{x}. \end{aligned}$$

(6.5)

Defining now $w(t):=u_{xx}(t, q(t,x_{0})),~~\frac{dq(t,x)}{dt}=-u_x(t,q(t,x))$, we obtain from the above inequality the relation

$$\begin{aligned} w_{t}(t, q(t,x_{0}))\ge \frac{1}{2}u^2_{xx}(t, q(t,x_{0}))-\frac{1}{2}u^2_{x}(t, q(t,x_{0})). \end{aligned}$$

(6.6)

Letting

$$\begin{aligned} V(t)&=(u_{x}-u_{xx})(t,q(t,x_0))=e^{-q(t,x_0)}\int _{-\infty }^{q(t, x_0)}e^{\xi }m_\xi (t,\xi )d\xi <0, \end{aligned}$$

(6.7)

$$\begin{aligned} K(t)&=(u_{x}+u_{xx})(t,q(t,x_0))=e^{q(t,x_0)}\int ^{\infty }_{q(t, x_0)}e^{-\xi }m_\xi (t,\xi )d\xi >0. \end{aligned}$$

(6.8)

Differentiating (6.7) with respect to x, we obtain

$$\begin{aligned} \frac{d}{dt}V(t) =e^{-q(t,x_0)}\int _{-\infty }^{q(t, x_0)}e^{\xi }m_{t\xi }(t,\xi )d\xi -\frac{d}{dt}q(t,x_0)V(t). \end{aligned}$$

(6.9)

Direct computations show that

$$\begin{aligned}&\int _{-\infty }^{q(t, x_0)}e^{\xi }m_{t\xi }(t,\xi )d\xi \\&\quad =\int _{-\infty }^{q(t, x_0)}e^{\xi }(u_{\xi }m_{\xi \xi }+2u_{\xi \xi }m_{\xi })d\xi \nonumber \\&\quad =\int _{-\infty }^{q(t, x_0)}e^{\xi }(\partial _{\xi }(u_{\xi }m_{\xi })+u_{\xi \xi }m_{\xi })d\xi \nonumber \\&\quad =\int _{-\infty }^{q(t, x_0)}e^{\xi }(-u_{\xi }m_{\xi }+u_{\xi \xi }m_{\xi })d\xi \nonumber \\&\quad =\int _{-\infty }^{q(t, x_0)}\partial _{\xi }\left[ e^{\xi } \left( -\frac{1}{2}u^2_{\xi \xi }+u_{\xi }u_{\xi \xi }\right) \right] d\xi - \int _{-\infty }^{q(t, x_0)}\left[ e^{\xi }\left( \frac{1}{2}u^2_{\xi \xi }+u^2_{\xi }\right) \right] d\xi \nonumber \\&\quad =e^{q(t,x_0)}\left[ -\frac{1}{2}u^2_{xx}(t,q(t,x_0))+u_{x}u_{xx}(t,q(t,x_0))\right] - \int _{-\infty }^{q(t, x_0)}\left[ e^{\xi }\left( \frac{1}{2}u^2_{\xi \xi }+u^2_{\xi }\right) \right] d\xi ,\nonumber \end{aligned}$$

(6.10)

which follow with (6.9) and $V(t)=(u_{x}-u_{xx})(t,q(t,x_0))$ yield

$$\begin{aligned} \frac{d}{dt}V(t) =&-\frac{1}{2}u^2_{xx}(t,q(t,x_0))+u_{x}u_{xx})(t,q(t,x_0))\nonumber \\&- \int _{-\infty }^{x}\left[ e^{\xi }\left( \frac{1}{2}u^2_{\xi \xi }+u^2_{\xi }\right) d\xi +u_{x}\right) (t,q(t,x_0))V(t) \nonumber \\ \le&-\frac{1}{2}u^2_{xx}(t,q(t,x_0))+\frac{1}{2}u^2_{x}(t,q(t,x_0))<0. \end{aligned}$$

(6.11)

In an analogous way,we obtain

$$\begin{aligned} \frac{d}{dt}K(t)&=\frac{1}{2}u^2_{xx}(t,q(t,x_0))+u_{x}u_{xx})(t,q(t,x_0))\nonumber \\&\quad + \int _{-\infty }^{x}\left[ e^{\xi }\left( \frac{1}{2}u^2_{\xi \xi }+u^2_{\xi }\right) d\xi -u_{x}\right) (t,q(t,x_0))K(t) \nonumber \\ {}&\ge \frac{1}{2}u^2_{xx}(t,q(t,x_0))-\frac{1}{2}u^2_{x}(t,q(t,x_0))>0. \end{aligned}$$

(6.12)

In view of (6.6)–(6.9) and (6.11)–(6.12), we see that

$$\begin{aligned} w_{t}(t, q(t,x_{0}))&\ge \frac{1}{2}u^2_{xx}(t, q(t,x_{0}))-\frac{1}{2}u^2_{x}(t, q(t,x_{0})) \ge -\frac{1}{2}V(t)K(t)\nonumber \\&\ge -\frac{1}{2}V(0)K(0)\ge 0. \end{aligned}$$

(6.13)

Assume that the solution exists globally in time. We now show that this leads to a contradiction.

From (6.13), by intergration, we see that

$$\begin{aligned} w(t)\ge w(0)-\frac{1}{2}V(0)K(0)t. \end{aligned}$$

(6.14)

Since $-V(0)W(0)>0$, and the $H^1({\mathbb {R}})-norm$ of $u_{x}$ is conservation law, there exists certainly some $t_0>0$ such that

$$\begin{aligned} w^2(t)\ge 2\Vert u_{x}\Vert _{L^\infty ({\mathbb {R}})},\ \ \ \ \ \ \ t>t_{0}. \end{aligned}$$

(6.15)

Combining (6.6) with (6.15) yields

$$\begin{aligned} \frac{d}{dt}w(t)\ge \frac{1}{4}w^2(t),\ \ \ \ \ \ \ t>t_{0}. \end{aligned}$$

(6.16)

By (6.7), $w(0)>0$, by (6.16), $w(t)>0, for\ \ \ t\ge 0.$ Then solving the inequality (6.16), we get

$$\begin{aligned} \frac{1}{w(t)}-\frac{1}{w(t_0)}+\frac{1}{4}(t-t_{0})\le 0. \end{aligned}$$

(6.17)

Taking into account that $\frac{1}{w(t)}>0$ and $\frac{1}{4}(t-t_{0})\rightarrow \infty$ as $t\rightarrow \infty$, we get a contradiction. This proves that the wave u(t, x) breaks in finite time. $\square$

Then we present another blow-up result.

Theorem 6.2

Assume that $\varepsilon>0,\ \ u_0\in H^s, ~~s>\frac{5}{2}$ and $m_{0}(x_0)<-(1+\varepsilon )[(8\Vert u'_{0}\Vert ^2_{H^1}\ln (1+\frac{2}{\varepsilon }) +\Vert u_{0}\Vert ^2_{H^2})^{\frac{1}{2}}+\Vert u_{0}\Vert _{H^2}]$, then the corresponding solution of (1.2) blows up in finite time.

Proof

By a standard density argument, here we may assume $s=3$ to prove the theorem.

Given $u_0 \in H^3$, let T be the maximal existence time of the corresponding solution to (1.1) with the initial data $u_0\in H^3$.

From (1.2), we obtain,

$$\begin{aligned} m_t-u_xm_x\nonumber =&-\frac{1}{2}m^2+\frac{1}{2}u_{x}^2-\frac{1}{2}u^2+um \\ \nonumber \le&-\frac{1}{2}m^2+\frac{1}{2}u_{x}^2-\frac{1}{2}u^2+\left( \frac{1}{4}m^2+u^2\right) \\ \nonumber \le&-\frac{1}{4}m^2+\frac{1}{2}u_{x}^2+\frac{1}{2}u^2 \\ \nonumber \le&-\frac{1}{4}m^2+\Vert u\Vert ^2_{H^2} \\ \le&-\frac{1}{4}m^2+(\Vert u_0\Vert _{H^{2}}+2\Vert u'_0\Vert ^2_{H^1}t)^2. \end{aligned}$$

(6.18)

Set now $w(t):=\inf _{x\in {\mathbb {R}}}[\frac{1}{4}m(t, q(t,x))],~~\frac{dq(t,x)}{dt}=-\frac{1}{4}u_x(t,q(t,x))$, fix $\varepsilon >0$. and take

$$\begin{aligned} T_{1}&=\frac{(8\Vert u'_{0}\Vert ^2_{H^1}\ln (1+\frac{2}{\varepsilon }) +\Vert u_{0}\Vert ^2_{H^2})^{\frac{1}{2}}-\Vert u_{0}\Vert _{H^2}}{4\Vert u'_{0}\Vert ^2_{H^1}}, \end{aligned}$$

(6.19)

$$\begin{aligned} K(T_{1})&=\frac{1}{2}\Vert u_{0}\Vert _{H^2}+\Vert u'_{0}\Vert ^2_{H^1}T_{1}, \end{aligned}$$

(6.20)

which satisfying

$$\begin{aligned}2K(T_{1})T_{1}-\ln \left( 1+\frac{2}{\varepsilon }\right) \ge 0, \end{aligned}$$

Then we obtain from the above inequality the relation

$$\begin{aligned} \frac{dw}{dt}\le -w^2+K^{2}(T_{1}),~~~~\forall t\in [0,T]\cap [0,T_{1}]. \end{aligned}$$

(6.21)

In view of $w(0)<-(1+\varepsilon )K_{T}$, we obtain

$$\begin{aligned} w(t)<-(1+\varepsilon )K(T_{1}), \quad \forall t\in [0,T]\cap [0,T_{1}]. \end{aligned}$$

By solving the inequality (6.21), we get

$$\begin{aligned} 0\ge \frac{2K(T_{1})}{w(t)-K(T_{1})} \ge -1+\frac{w(0)+K(T_{1})}{w(0)-K(T_{1})}e^{2tK(T_{1})}. \end{aligned}$$

(6.22)

Since

$$\begin{aligned} 0<\frac{w(0)-K(T_{1})}{w(0)+K(T_{1})}=1-\frac{2K(T_{1})}{w(0)+K(T_{1})}\le 1+\frac{2}{\varepsilon }. \end{aligned}$$

(6.23)

then it follows that there exists $T^*$

$$\begin{aligned} 0<T^*\le \frac{1}{2K(T_{1})}\ln \frac{w(0)-K(T_{1})}{w(0)+K(T_{1})} \le \frac{1}{2K(T_{1})}\ln {\left( 1+\frac{2}{\varepsilon }\right) }\le T_{1}, \end{aligned}$$

(6.24)

such that

$$\begin{aligned} w(t)\le -K(T_{1})+\frac{2K(T_{1})}{1-\frac{w(0)-K(T_{1})}{w(0)+K(T_{1})}e^{-2K(T_{1})t}}\rightarrow -\infty , \end{aligned}$$

(6.25)

as $t\rightarrow T^{\star }$. This proves that the wave u(t, x) breaks in finite time. $\square$

Remark 6.3

The Eq. (1.2) and the CH equation are truly different for the blow-up phenomena. The solution to the CH equation with the particular condition will blow up in finite time. Similarly, the solution to (1.2) will blow up in finite time with the corresponding condition. However, the corresponding solution of (1.2) with $m_{0}(x_0)<-(1+\varepsilon )[(8\Vert u'_{0}\Vert ^2_{H^1}\ln (1+\frac{2}{\varepsilon }) +\Vert u_{0}\Vert ^2_{H^2})^{\frac{1}{2}}+\Vert u_{0}\Vert _{H^2}]$ blows up in finite time. But we can’t deduce the blow up phenomena to the CH equation with the corresponding condition.

Finally, we prove the exact blow-up rate for blowing-up solutions m(t, x) to (1.2) guaranteed by Theorem 6.2. In order to establish this result, we need the following useful lemma.

Lemma 6.4

[13] Let $T > 0$ and $u \in C^1([0, T );H^2)$. Then for every $t \in [0, T )$, there exists at least one point $\xi (t) \in {\mathbb {R}}$ with

$$\begin{aligned} a(t) \triangleq \sup _{x\in {\mathbb {R}}} (v_{x}(t, x)) = v_{x}(t, \xi (t)). \end{aligned}$$

The function a(t) is absolutely continuous on (0, T) with

$$\begin{aligned} \frac{da}{dt}= v_{tx}(t, \xi (t)) \quad a.e. ~on ~~~(0, T ). \end{aligned}$$

Theorem 6.5

Let $u_0\in H^s,~s>\frac{5}{2}$, $m_{0}(x_0)<-(\Vert u_0\Vert _{H^{2}}+\Vert u_0\Vert ^2_{H^2}T)$ and T be the blow-up time of the corresponding solution u to (1.2). Then

$$\begin{aligned} \lim _{t\rightarrow T}\left( \sup _{x\in {\mathbb {R}}}[m(t,x)](T-t)\right) =2. \end{aligned}$$

(6.26)

Proof

As mentioned earlier, we only need to prove the theorem for $s=3.$

Note that $G(x)=\frac{1}{2}e^{-|x|}$ and $G(x)\star f =(1-\partial _x^2)^{-1}f$ for all $f \in L^2(R)$ and $G \star m=u$. Then we can rewrite (1.2) as follows:

$$\begin{aligned} u_t=\frac{1}{2}u^2_{x}+G\star \left[ u_{x}^2+\frac{1}{2} u_{xx}^{2}\right] . \end{aligned}$$

(6.27)

Defining now $\frac{dq(t,x)}{dt}=-u_x(t,q(t,x))$. In view of (1.2) and (6.27), we obtain

$$\begin{aligned} \frac{d(m-u)(t,q(t,x))}{dt}+\frac{1}{2}(m-u)^2=-u_{x}^2-G(x)*\left( u^2_{x}+\frac{1}{2}u^2_{xx}\right) . \end{aligned}$$

(6.28)

Thanks to (6.28), Lemmas 4.1, we have

$$\begin{aligned} \left| \frac{d(m-u)(t,q(t,x))}{dt}+\frac{1}{2}(m-u)^2\right|&=u_{x}^2+G(x)*\left( u^2_{x}+\frac{1}{2}u^2_{xx}\right) ,\\&\le 2\Vert u_{x}\Vert _{H^1}^{2}\\&\le 2\Vert u'_{0}\Vert _{H^1}^2. \end{aligned}$$

Defining now $w(t):=\inf _{x\in {\mathbb {R}}}[\frac{1}{2}(m-u)(t, q(t,x))]$, we obtain from the above inequality the relation

$$\begin{aligned} \left| \frac{dw}{dt}+w^2\right| \le 2\Vert u'_{0}\Vert _{H^1}^2,~~~~\forall t\in (0,T). \end{aligned}$$

(6.29)

For every $\varepsilon \in (0, \frac{1}{2})$, in view of (6.25), we can find a $t_0\in (0,T)$ such that

$$\begin{aligned} w(t_0)<-\sqrt{2\Vert u'_{0}\Vert _{H^1}^2+\frac{2\Vert u'_{0}\Vert _{H^1}^2}{\varepsilon }} <-\sqrt{2}\Vert u'_{0}\Vert _{H^1}. \end{aligned}$$

Thanks to (6.25) and (6.29), we have $w(t)<-\mathbf {\widetilde{C_{T}}}.$ This implies that w(t) is decreasing on $[t_0,T),$ hence,

$$\begin{aligned} w(t)<-\sqrt{2\Vert u'_{0}\Vert _{H^1}^2+\frac{2\Vert u'_{0}\Vert _{H^1}^2}{\varepsilon }} <-\sqrt{\frac{2\Vert u'_{0}\Vert _{H^1}^2}{\varepsilon }},~~~~\forall t\in [t_0,T). \end{aligned}$$

Noticing that $-w^2-2\Vert u'_{0}\Vert _{H^1}^2 \le \frac{dw(t)}{dt}\le -w^2+ 2\Vert u'_{0}\Vert _{H^1}^2,~~~a.e.~t\in (t_0,T),$ we get

$$\begin{aligned} -1-\varepsilon \le \frac{d}{dt}\left( -\frac{1}{w(t)}\right) \le -1+\varepsilon ,~~~a.e.~t\in (t_0,T). \end{aligned}$$

(6.30)

Integrating (6.30) with respect to $t\in [t_0, T)$ on (t, T) and applying $\lim _{t\rightarrow T}w(t)=-\infty$ again, we deduce that

$$\begin{aligned} (-1-\varepsilon )(T-t)\le \frac{1}{w(t)}\le (-1+\varepsilon )(T-t). \end{aligned}$$

(6.31)

Since $\varepsilon \in (0,\frac{1}{2})$ is arbitrary, it then follows from (6.31) that (6.26) holds. Noting that Lemma 4.2, we get $\lim _{t\rightarrow T}(\inf _{x\in {\mathbb {R}}}u(t,x)(T-t))=0.$

This completes the proof of the theorem. $\square$