\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 44, pp. 1--27.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2013/44\hfil Wave-breaking phenomena] {Wave-breaking phenomena and global solutions for periodic two-component Dullin-Gottwald-Holm systems} \author[M. Zhu, J. Xu \hfil EJDE-2013/44\hfilneg] {Min Zhu, Junxiang Xu} \address{Min Zhu \newline Department of Mathematics, Nanjing Forestry University \\ Nanjing 210037, China} \email{zhumin@njfu.edu.cn} \address{Junxiang Xu \newline Department of Mathematics, Southeast University \\ Nanjing 211189, China} \email{xujun@seu.edu.cn} \thanks{Submitted November 14, 2012. Published February 8, 2013.} \subjclass[2000]{35B30, 35G25} \keywords{Two-component Dullin-Gottwald-Holm system; \hfill\break\indent periodic two-component $b$-family system; blow-up; wave-breaking; global solution} \begin{abstract} In this article we study the initial-value problem for the periodic two-component $b$-family system, including a special case, when $b = 2$, which is referred to as the two-component Dullin-Gottwald-Holm (DGH) system. We first show that the two-component $b$-family system can be derived from the theory of shallow-water waves moving over a linear shear flow. Then we establish several results of blow-up solutions corresponding to only wave breaking with certain initial profiles for the periodic two-component DGH system. Moreover, we determine the exact blow-up rate and lower bound of the lifespan for the system. Finally, we give a sufficient condition for the existence of the strong global solution to the periodic two-component DGH system. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In recent years, Degasperis, Holm and Hone \cite {DPDH} (see also \cite{HoSt}) studied the following nonlinear $b$-family equation (up to a rescaling, shift and Galilean's transformation), \begin{equation} \label{e1.1} m_t-Au_x+um_x+bu_xm+\gamma{u_{xxx}}=0, \quad x \in\mathbb{R}, \; t>0, \end{equation} where $m=u-{\alpha^2}u_{xx}. $ One can rewrite equation \eqref{e1.1} in terms of $u(x,t)$ as follows: \begin{equation} \label{e1.2} u_t-{\alpha^2}u_{xxt}-Au_x+(b+1)uu_x+\gamma{u_{xxx}} ={\alpha^2}(bu_xu_{xx}+uu_{xxx}), \quad x \in\mathbb{R}, \; t>0. \end{equation} This equation can be regarded as a model of water waves by using asymptotic expansions directly in the Hamiltonian for Euler's equation in the shallow water regime \cite{DuGoHo, HoSt}, where $u(t,x)$ stands for the horizontal velocity of the fluid, $ m $ is the momentum density, and $A$ is a nonnegative parameter related to the critical shallow water speed. The real dimensionless constant $b$ is a parameter which provides the competition, or balance, in fluid convection between nonlinear steepening and amplification due to stretching, it is also the number of covariant dimensions associated with the momentum density $ m$. It is believed that the Korteweg-de Vries (KdV) equation ($\alpha=0$ and $b=2$), the Camassa-Holm (CH) equation ($ b =2$) \cite {CaHo, FoFu} (when $ b =2 $ and $ \gamma \ne 0$, it is also referred to as the Dullin-Gottwald-Holm (DGH) equation \cite{CaHo, DuGoHo}), and the Degasperis-Procesi (DP) equation ($ b =3 $) \cite{DP} are the only three integrable equations in the $b$-family equation \eqref{e1.2} \cite {DuGoHo, DuGoHo1, DPDH, DP, HoSt, Iv1}. When $ A= \gamma= 0$, \eqref{e1.2} admits not only the peakon solutions for any $b$ of the form $u(t,x)=ce^{-|x-ct|}$, $c \in \mathbb{R}$, but also multipeakon solutions \cite {BeSaSz, DPDH, HoSt} (see also \cite{CaHoTi} for the case of existence of infinite many peakons) defined by $$ u(x, t) = \sum_{j = 1}^{N} p_j(t) e^{-|x - q_j(t)|}, $$ where the canonical positions $q_j$ and momenta $p_j$ (with $j =1,\dots, N$) satisfy the following system of ordinary differential equations with discontinuous right-hand side. $$ p_j'=(b-1)\sum_{k=1}^{N}p_jp_k sgn(q_j-q_k)e^{-|q_j-q_k|} $$ and $$ q_j'=\sum_{k=1}^{N}p_ke^{-|q_j-q_k|}. $$ If $\alpha=0$ and $b=2$, equation \eqref{e1.2} becomes the well-known KdV equation which describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity. Its solitary waves are solitons. The Cauchy problem of the KdV equation has been the subject of a number of studies, and a satisfactory local or global existence theory is now in hand \cite{Tao}. It is observed that the KdV equation does not accommodate wave breaking (by wave breaking we understand that the wave profile remains bounded while its slope becomes unbounded in finite time \cite {Wh}). When $b=2$ and $\gamma=0$, equation \eqref{e1.2} recovers the standard CH equation, modeling the unidirectional propagation of shallow water waves over a flat bottom \cite{CaHo, CoLa, FoFu}. The CH equation is also a model for the propagation of axially symmetric waves in the hyperelastic rods \cite{Dai}. Its solitary waves are smooth if $A>0$ and peaked in the limiting case $A=0$ \cite{CaHo, CaHoHy, CaHoTi}. Recently, it was claimed in \cite{LM} that the CH equation might be relevant to the modeling of tsunami. If $b=3$ and $A=\gamma=0$ in equation \eqref{e1.2}, then it recovers the DP equation. The DP equation can be also regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as the CH equation \cite {CoLa}. The formal integrability of the DP equation was obtained in \cite{DPDH} by constructing a Lax pair. It has a bi-Hamiltonian structure. The DP equation has not only peaked solitons and periodic peaked solitons, but also shock peakons \cite{Lu} and the periodic shock waves \cite{EsYue}. The CH and DP equations have global strong solutions and also blow-up solutions in finite time, for instance, see \cite{Co1, CoEs1, CoEs3, C-Mc, EsYue, Liu, LiYi, LiYi2} and references therein, with a different class of initial profiles in the Sobolev spaces $ H^s(\mathbb{R}), \ s > 3/2. $ It is shown in \cite{BrCo1} and \cite{BrCo2} that solutions of the CH equation can be uniquely continued after breaking as either global conservative or global dissipative weak solutions. The advantage of the CH and DP equations in comparison with the KdV equation lies in the fact that the CH and DP equations have peaked solitons and models wave breaking. Wave breaking is one of the most intriguing long-standing problems of water wave theory \cite{Wh}. The peaked solitons are the presence of solutions in the form of peaked solitary waves or "peakons" \cite{CaHo, CaHoHy, CaHoTi, DP} $u(t,x)=ce^{-|x-ct|},c\neq0$, which are smooth except at the crests, where they are continuous, but have a jump discontinuity in the first derivative. The peakons replicate a feature that is characteristic for the waves of great height-waves of the largest amplitude that are exact solutions of the governing equations for water waves \cite{Co2, Toland, CoEs4}. These peakons are shown to be stable \cite {CoMo, CoSt, LiLi}. The interest in the $b$-family equation inspired the search for various generalizations of this equation. The following two-component integrable Camassa-Holm system was first derived in \cite{OlRo} and can be viewed as a model in the context of shallow water theory \cite{CoIv, Iv}, \begin{equation} \label{e1.3} \begin{gathered} m_t-Au_x+um_x+ 2u_xm+\rho\rho_x=0,\\ m=u-u_{xx},\\ \rho_t+(u\rho)_x=0, \end{gathered} \end{equation} where $\rho(t,x)$ is related to the free surface elevation from equilibrium(or scalar density), and the parameter $A$ characterizes a linear underlying shear flow. Obviously, if $\rho=0$, then \eqref{e1.3} becomes the CH equation. Many recent works are devoted in studying system \eqref{e1.3} (see, for instance, \cite{CoIv,EsLeYi,FuLiQu,GuYi1, GuYi2, GuLi1,GuLi2, Iv, ZhLi} and references therein). In the presence of a linear shear flow and nonzero vorticity, we will follow Ivanov's approach \cite{Iv} to derive the following two-component $b$-family system with any $ b \ne -1$. \begin{equation} \label{e1.4} \begin{gathered} m_t-Au_x+um_x+bu_xm+\gamma u_{xxx}+\rho\rho_x=0,\\ m=u-u_{xx},\\ \rho_t+(u\rho)_x=0. \end{gathered} \end{equation} Note when $\rho=0$, we recover the $b$-family equation \eqref{e1.1}. In terms of $u$ and $\rho$, we obtain the equivalent form of system \eqref{e1.4}; that is, \begin{equation} \label{e1.5} \begin{gathered} u_t-u_{txx}-Au_x+(b+1)uu_x-bu_xu_{xx}-uu_{xxx}+\gamma u_{xxx}+\rho\rho_x=0,\\ \rho_t+(u\rho)_x=0,\\ \end{gathered} \end{equation} with the boundary assumptions $u\to 0$ and $\rho\to 1$ as $|x|\to\infty$. Note that when $ b= 2$, equation\eqref{e1.5} is the two-component Camassa-Holm system, which has the bi-Hamiltonian structure and complete integrability via the inverse scattering transform method. It can be written as compatibility conditions of two linear systems (Lax pair) with a spectral parameter $\xi$, that is \begin{gather*} \Psi_{xx}= \Big( -\xi^2\rho^2+\xi \Big(m-\frac{A}{2}+\frac{\gamma}{2} \Big) +\frac{1}{4} \Big) \Psi,\\ \Psi_t= \Big(\frac{1}{2\xi}-u+\gamma \Big)\Psi_x+\frac{1}{2}u_x\Psi. \end{gather*} Moreover, this system has the following two Hamiltonians $$ E(u,\rho)=\frac{1}{2}\int \left ( u^2+u_x^2+(\rho-1)^2 \right ) dx $$ and $$ F(u,\rho)=\frac{1}{2}\int \left ( u^3+uu_x^2-Au^2-\gamma u_x^2+2u(\rho-1)+u(\rho-1)^2 \right ) dx. $$ The goal of this article is to study the initial-value problem for the periodic two-component $b$-family system, including a special case, $b = 2$, which is the two-component DGH system. We first derive the two-component $b$-family system from the shallow-water wave theory. Then we establish several results of blow-up solutions corresponding to only wave breaking with certain initial profiles for the periodic two-component DGH system. The difficulty to deal with blow-up solutions is that there is no uniform characteristics for this system. In this case, we make use of the different diffeomorphism of the trajectory $q_2$ defined in \eqref{e4.4}, which captures the maximum/minimum of $u_x$. Therefore the transport equation for $\rho$ can coincide with the equation for $u$. The rest of this paper is organized as follows. In Section 2, we follow the modeling approach in \cite{Iv} to derive the two-component $b$-family system. Then applying Kato's semigroup theory, we establish the result of local well-posedness for the two component $b$-family system in Section 3. In Section 4, we analyze the wave-breaking phenomenon of the periodic two-component DGH system and give the precise blow-up scenarios and several wave-breaking data. In addition, we determine the blow-up rate and low bound of the lifespan. In the last section, we provide a sufficient condition for the existence of global solution. \noindent {\it Notation}. Throughout this paper, we identity periodic functions with function spaces over the unit circle $\mathbb{S}$ in $\mathbb{R}^2$, i.e. $\mathbb{S}=\mathbb{R}/\mathbb{Z}$. \section{Derivation of the model}\label{sec_model} Following Ivanov's approach in \cite{Iv} , we consider the motion of an inviscid incompressible fluid with a constant density $\varrho$ governed by the Euler equations \begin{gather*} \vec{v}_t+\left( \vec{v}\cdot\nabla \right)\vec{v}=-{1\over\varrho}\nabla P+\vec{g},\\ \nabla\cdot \vec{v}=0, \end{gather*} where $\vec{v}(t,x,y,z)$ is the velocity of the fluid, $P(t,x,y,z)$ is the pressure and $\vec{g}=(0,0,-g)$ is the gravity acceleration. Using the shallow water approximation and non-dimensionalization, the above equations can be written as \begin{gather*} u_t+\varepsilon(uu_x+wu_z)=-p_x,\\ \delta^2\left( w_t+\varepsilon(uw_x+ww_z) \right)=-p_z,\\ u_x+w_z=0,\\ w=\eta_t+\varepsilon u\eta_x,\quad p=\eta \quad \hbox{on } z=1+\varepsilon\eta,\\ w=0\quad\hbox{on } z=0, \end{gather*} where $\vec{v}=(u,0,w)$ and $p(x,z,t)$ is the pressure variable measuring the deviation from the hydrostatic pressure distribution and $\eta(t,x)$ is the deviation from the mean level $z=h$ of the water surface. $\varepsilon=a/h$ and $\delta=h/\lambda$ are the two dimensionless parameters with $a$ being the typical amplitude of the wave and $\lambda$ being the typical wavelength of the wave. In the presence of an underlying shear flow, the horizontal velocity of the flow becomes $u+\tilde{U}(z)$. We take the simplest case $\tilde{U}(z)=Az$ in which $A>0$ is a constant. Notice that the Burns condition gives the shallow-water limit of the dispersion relation for the waves with vorticity , hence determines the speed of propagation of the linear waves. From Burns condition \cite{CoVa, Jo} one has the following expression for the speed $c$ of the traveling waves in linear approximation, \begin{equation} \label{e2.1} %\label{defnc} c={1\over2}\left( A\pm\sqrt{4+A^2} \right). \end{equation} In the case of the constant vorticity $\omega=A$, we obtain the following equations for $u_0$ and $\eta$ by ignoring the terms of $O(\varepsilon^2,\delta^4,\varepsilon\delta^2)$ \begin{gather} \Big( u_0-{1\over2}\delta^2 u_{0,xx} \Big)_t+\varepsilon u_0u_{0,x}+\eta_x-{A\over3}\delta^2 u_{0,xxx}=0, \label{eqnu_0}\\ \eta_t+A\eta_x+\Big((1+\varepsilon\eta)u_0+{A\over2}\varepsilon \eta^2 \Big)_x-{1\over6}\delta^2 u_{0,xxx}=0, \label{eqneta} \end{gather} where $u_0$ is the leading order approximation for $u$ (see the details in \cite{Iv}). Let both of the parameters $ \epsilon $ and $\delta $ go to $0$. Then by \eqref{eqnu_0} and \eqref{eqneta}, we have the system of linear equations \begin{gather*} u_{0,t}+\eta_x=0,\\ \eta_t+A\eta_x+u_{0,x}=0. \end{gather*} This in turn implies that $\eta_{tt}+A\eta_{tx}-\eta_{xx}=0$. Introducing a new variable \begin{equation*} \rho=1+\varepsilon\alpha\eta+\varepsilon^2\beta\eta^2+\varepsilon\delta^2\mu u_{0,xx}, \end{equation*} for some constants $\alpha,\beta$ and $\mu$ satisfying \begin{gather*} {\mu\over\alpha}={1\over6(c-A)},\\ \alpha=1+{Ac\over2}+{\beta\over\alpha}, \end{gather*} then equations \eqref{eqnu_0} and \eqref{eqneta} become \begin{equation} \label{CH2full} \begin{gathered} m_t+Am_x-Au_{0,x}-{1\over{6c^2(c-A)}}\delta^2u_{0,xxx}\\ + \varepsilon\Big(1- \frac{\alpha^2+2\beta}{\alpha}c^2\Big)u_0u_{0,x} +\frac{1}{2\varepsilon\alpha}(\rho^2)_x=0,\\ \rho_t+A\rho_x+\alpha\varepsilon(\rho u_0)_x=0, \end{gathered} \end{equation} where $ m=u_0-{\frac12}\delta^2u_{0,xx}$. Since $ b \ne -1 $ and \begin{equation*} (b+1)u_0u_{0,x}=bmu_{0,x}+u_0m_x+O(\delta^2), \end{equation*} equation \eqref{CH2full} can be reformulated at the order of $O(\varepsilon,\delta^2)$ as \begin{align*} & m_t+Am_x-Au_{0,x}-{1\over{6c^2(c-A)}}\delta^2u_{0,xxx}\\ &+ {\varepsilon\over{b+1}}\Big(1- \frac{\alpha^2+2\beta}{\alpha}c^2\Big)(bmu_{0,x}+u_0m_x) +\frac{1}{2\varepsilon\alpha}(\rho^2)_x=0. \end{align*} Using the scaling $ u_0\to {1\over\alpha\varepsilon}u_0,\ x\to{\delta}x$ and $\ t\to{\delta}t$, then \eqref{CH2full} becomes \begin{gather*} m_t+Am_x-Au_{0,x}-{1\over{6c^2(c-A)}}u_{0,xxx}+\\ {1\over(b+1)\alpha}\Big(1- \frac{\alpha^2+2\beta}{\alpha}c^2\Big)(bmu_{0,x}+u_0m_x)+\frac{1}{2}(\rho^2)_x=0,\\ m=u_0-u_{0,xx},\\ \rho_t+A\rho_x+(\rho u_0)_x=0. \end{gather*} Now if we choose \begin{equation*} \frac{1}{(b+1)\alpha}\Big(1- \frac{\alpha^2+2\beta}{\alpha}c^2\Big)=1 \end{equation*} and denote $ \gamma=-{1\over{6c^2(c-A)}}$, then we arrive at \begin{equation} \label{genCH2_1} \begin{gathered} m_t+Am_x-Au_{0,x}+bmu_{0,x}+u_0m_x+\gamma{u_{0,xxx}}+\rho\rho_x=0,\\ m=u_0-u_{0,xx},\\ \rho_t+A\rho_x+(\rho u_0)_x=0. \end{gathered} \end{equation} Thus the constants $\alpha, \beta, \mu$ and $c$ satisfy \begin{gather*} \alpha=\frac{c^2(c^2+1)+1}{3c^2+b+1},\quad \beta=\alpha^2-\alpha\Big(1+\frac{Ac}{2}\Big),\\ \mu=\frac{\alpha}{6(c-A)},\quad c^2-Ac-1=0. \end{gather*} With a further Galilean transformation $x\to x-ct$, $t\to t$, we can drop the terms $A\rho_x$ and $ A m_x $ in \eqref{genCH2_1} and obtain the two-component $b$-family system \eqref{e1.4} or \eqref{e1.5}. \section{Local well-posedness} In this section, we will apply Kato's semigroup theory to establish the local well-posedness for the following periodic initial-value problem to \eqref{e1.5}. \begin{equation} \label{e3.1} \begin{gathered} u_t+(u-\gamma)u_x=-\partial_x (1-\partial^2_x)^{-1} \Big({b\over2}u^2+{{3-b}\over2}u^2_x+(\gamma-A)u+{1\over2}\rho^2\Big),\\ t\geq0,\; x\in \mathbb{R},\\ \rho_t+(u\rho)_x=0,\quad t\geq0,\; x\in \mathbb{R},\\ u(0,x)=u_0(x), \quad x \in\mathbb{R},\\ \rho(0,x)=\rho_0(x), \quad x\in \mathbb{R},\\ u(t,x+1)=u(t,x), \quad t\ge0,\; x\in \mathbb{R},\\ \rho(t,x+1)=\rho(t,x), \quad t\ge0,\; x\in \mathbb{R}. \end{gathered} \end{equation} For convenience, we present here Kato's theorem in a form suitable for our purpose. Consider the abstract quasilinear evolution equation \begin{equation} \label{e3.2} \begin{gathered} \frac {dv}{dt} + A(v) v =f(v) \quad t\geq0,\\ v(0)= v_0. \end{gathered} \end{equation} Let $X$ and $Y$ be two Hilbert spaces such that $ Y$ is continuously and densely embedded in $X$ and let $Q: Y\to X$ be a topological isomorphism. Let $L(Y,X)$ denote the space of all bounded linear operators from $Y$ to $X$, particularly, it is denoted by $L(X)$ if $X=Y$. The linear operator $A$ belongs to $G(X, 1, \beta)$ where $\beta$ is a real number, if $-A$ generates a $C_0$-semigroup such that $\|e^{-sA}\| _{L(X)} \leq e^{\beta s}$. We make the following assumptions, where $\mu_i (1=1,2,3,4)$ are constants depending only on $\max \{ \|y\|_Y,\ \|z\|_Y\}$: \noindent (i) $A(y) \in L(Y,X)$ for $y\in Y$ with \[ \| \big( A(y)-A(z) \big)w \|_X \leq \mu_1 \|y-z\|_X \|w\|_Y,\quad y,z, w\in Y \] and $A(y) \in G(X, 1, \beta)$ (i.e., $A(y)$ is quasi-m-accretive), uniformly on bounded sets in $Y$. \noindent (ii) $QA(y)Q^{-1} = A(y) + B(y)$, where $B(y) \in L(X)$ is bounded, uniformly on bounded sets in $Y$. Moreover, \[ \| \big( B(y) - B(z) \big) w\|_X \leq \mu_2 \|y-z\|_Y \|w\|_X,\quad y,z\in Y, \; w\in X. \] \noindent (iii) $f: Y\to Y$ extends to a map from $X$ into $X$, is bounded on bounded sets in $Y$, and satisfies \[ \| f(y) - f(z) \|_Y \leq \mu_3 \|y-z\|_Y, \quad y, z\in Y \] and \[ \| f(y) - f(z) \|_X \leq \mu_4 \|y-z\|_X, \quad y, z\in Y. \] \begin{lemma}[\cite{Ka1}] \label{lem3.1} Assume conditions {\rm (i), (ii) (iii)} hold. Given $v_0 \in Y$, there is a maximal $T>0$ depending only on $\|v_0\|_Y$ and a unique solution $v$ to \eqref{e3.2} such that \[ v= v(\cdot, v_0) \in C \big( [0,T); Y\big) \cap C^1 \big([0,T); X\big). \] Moreover, the map $v_0 \mapsto v(\cdot, v_0)$ is a continuous map from $Y$ to $C \big( [0,T); Y\big) \cap C^1 \big([0,T); X\big)$. \end{lemma} We now provide the framework in which we shall reformulate problem \eqref{e3.1}. \begin{theorem} \label{thm3.2} Given an initial data $(u_0, \rho_0) \in H^s (\mathbb{S}) \times H^{s-1} (\mathbb{S}),\ s\geq 2$, there exists a maximal $T = T\left(\| (u_0, \rho_0)\|_{H^s(\mathbb{S}) \times H^{s-1}(\mathbb{S}) } \right) >0$ and a unique solution \[(u,\rho) \in C \left( [0,T); H^s(\mathbb{S}) \times H^{s-1}(\mathbb{S}) \right) \cap C^1 \left( [0,T); H^{s-1}(\mathbb{S})\times H^{s-2}(\mathbb{S}) \right) \] of system \eqref{e3.1}. Moreover, the solution $(u,\rho)$ depends continuously on the initial value $(u_0, \rho_0)$ and the maximal time of existence $T>0$ is independent of $s$. \end{theorem} The remaining of this section is devoted to the proof of Theorem \ref{thm3.2}. Let \begin{gather} U = \begin{pmatrix} u \\ \rho \end{pmatrix}, \nonumber\\ \label{e3.3} A(U)= \begin{pmatrix} (u - \gamma ) \partial_x & 0 \\ 0 & u\partial_x \end{pmatrix}\\ \label{e3.4} f(U) = \begin{pmatrix} -\partial_x ( 1 - \partial_x^2)^{-1} \left( {b\over2}u^2+{{3-b}\over2}u^2_x+(\gamma-A)u+{1\over2}\rho ^2 \right)\\ -u_x \rho \end{pmatrix} \end{gather} $ Y=H^s \times H^{s-1}$, $X=H^{s-1} \times H^{s-2}$, $\Lambda = (1-\partial_x^2)^{1/2}$ and $$ Q = \begin{pmatrix} \Lambda & 0 \\ 0 & \Lambda \end{pmatrix} . $$ Obviously, $Q$ is an isomorphism of $H^s \times H^{s-1}$ onto $H^{s-1} \times H^{s-2}$. Thus, to derive Theorem \ref{thm3.2}, we only need to check that $A(U)$ and $f(U)$ satisfy the conditions (i)-(iii), and this can be formulated through several lemmas. The following lemmas from \cite{Ka1} and \cite{Ka2} are useful in our proofs. \begin{lemma}[\cite{Ka1}] \label{lem3.3} Let $r, t$ be two real numbers such that $-r \frac12 $$ and $$ \|fg\|_{H^{r+t- \frac12}} \leq c \|f\|_{H^r} \|g\|_{H^t},\quad if\ r<\frac12, $$ where $c$ is a positive constant depending on $r$ and $t$. \end{lemma} \begin{lemma}[\cite{Ka2}] \label{lem3.4} Let $f\in H^r$ for some $r>\frac32$. Then $$ \| \Lambda ^{-\bar s} [ \Lambda^{\bar s + \bar t +1}, M_f ] \Lambda^{- \bar t} \|_{L(L^2)} \leq c \|\partial _x f\|_{r-1}, \ \ |\bar s|, \ |\bar t| \leq r-1, $$ where $M_f$ is the operator of multiplication by $f$ and $c$ is a constant depending only on $\bar s$ and $\bar t$. \end{lemma} \begin{lemma} \label{lem3.5} With $U \in H^s (\mathbb{S}) \times H^{s-1}(\mathbb{S}) (s\geq 2)$, the operator $A(U)$ belongs to $G(H^{s-1}(\mathbb{S}) \times H^{s-2}(\mathbb{S}),1,\beta)$. \end{lemma} \begin{proof} Taking the $H^{s-1} \times H^{s-2}$ inner product with $W=\begin{pmatrix} w_1 \\ w_2 \end{pmatrix}$ on both sides of the equation \[ \frac {dW}{dt} + A(U) W =0 \] gives \begin{align*} &\frac12 \frac {d}{dt} \|W\|^2_{H^{s-1} \times H^{s-2}}\\ &= -\langle W, A(U) W \rangle_{(s-1)\times (s-2)}\\ &= - \Big\langle \begin{pmatrix} w_1 \\ w_2 \end{pmatrix}, \begin{pmatrix} (u- \gamma) \partial_x w_1 \\ u \partial_x w_2 \end{pmatrix} \Big\rangle _{(s-1) \times (s-2)}\\ &= -\langle w_1, (u-\gamma) \partial_x w_1 \rangle_{s-1} - \langle w_2, u\partial w_2 \rangle _{s-2}\\ &= - \langle \Lambda ^{s-1} w_1, \Lambda ^{s-1} \big( ( u -\gamma )\partial_x w_1 \big) \rangle - \langle \Lambda ^{s-2} w_2, \Lambda ^{s-2} \big( u\partial_x w_2 \big) \rangle\\ &= -\langle \Lambda ^{s-1} w_1, [\Lambda ^{s-1}, u-\gamma ] \partial_x w_1 \rangle - \langle \Lambda ^{s-1} w_1, (u-\gamma) \partial_x \Lambda ^{s-1} w_1 \rangle\\ &\quad -\langle \Lambda ^{s-2} w_2, [\Lambda ^{s-2}, u] \partial_x w_2 \rangle - \langle \Lambda ^{s-2} w_2, u \partial_x \Lambda ^{s-2} w_2 \rangle\\ &= -\langle \Lambda ^{s-1} w_1, [\Lambda ^{s-1}, u-\gamma ] \partial_x w_1 \rangle - \frac12 \langle \Lambda ^{s-1} w_1, u_x \partial_x \Lambda ^{s-1} w_1 \rangle\\ & \quad -\langle \Lambda ^{s-2} w_2, [\Lambda ^{s-2}, u] \partial_x w_2 \rangle -\frac12 \langle \Lambda ^{s-2} w_2, \partial_x u \Lambda ^{s-2} w_2 \rangle\\ &\leq \| \Lambda ^{s-1} w_1 \|_{L^2}^2 \| [ \Lambda ^{s-1}, u-\gamma ] \Lambda ^{2-s} \|_{L(L^2)} + \frac12 \|u_x\| _{L^{\infty}} \|\Lambda ^{s-1} w_1 \|_{L^2}\\ & \quad + \| \Lambda ^{s-2} w_2 \|_{L^2}^2 \| [ \Lambda ^{s-2}, u ] \Lambda ^{3-s} \|_{L(L^2)} + \frac12 \|u_x\| _{L^{\infty}} \|\Lambda ^{s-2} w_2 \|_{L^2}\\ &\leq c \left( \|U\|_{H^s} + |\gamma| \right) \left( \|w_1\|^2_{H^{s-1}} + \|w_2\|^2_{H^{s-2}}\right) \\ &= c \left( \|U\|_{H^s} + |\gamma| \right) \|W \|^2_{H^{s-1} \times H^{s-2}}, \end{align*} where use has been made of Lemma \ref{lem3.4} with $r=0, \bar t=s-2$ and $\bar s=0$, $\bar t=s-3$, respectively. By integrating both of sides in the above the estimate, it follows that $A(U) \in G \left(H^{s-1}(\mathbb{S}) \times H^{s-2}(\mathbb{S}), 1, c (\|u\|_{H^s} + \gamma) \right)$ \end{proof} \begin{lemma} \label{lem3.6} The operator $A(U)$ defined by \eqref{e3.3} belongs to t$L( H^s\times H^{s-1}, H^{s-1}\times H^{s-2})$. Moreover \begin{equation} \label{e3.5} \begin{gathered} \|\left( A(U) - A(V) \right) W \|_{H^{s-1} \times H^{s-2}} \leq \mu_1 \|U-V\|_{H^s \times H^{s-1}} \|W\|_{H^s \times H^{s-1}}, \\ U, V,W \in {H^s \times H^{s-1}}. \end{gathered} \end{equation} \end{lemma} \begin{proof} In view of \eqref{e3.3}, we have \begin{align*} \left( A(U) - A(V) \right) W & = \begin{pmatrix} (u - \gamma ) \partial_x - (v_1 - \gamma ) \partial_x & 0 \\ 0 & u\partial_x - v_1 \partial_x \end{pmatrix} \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} \\ & = \begin{pmatrix} (u - v_1) \partial_x w_1 \\ (u - v_1) \partial_x w_2 \end{pmatrix}. \end{align*} Since $H^{s-1}\ (s\geq 2)$ is a Banach algebra, taking $r=s-1$, $t=s-2$ in Lemma \ref{lem3.3}, we have \begin{align*} &\left\| \left( A(U) - A(V) \right) W \right\|_{H^{s-1} \times H^{s-2}} \\ &\leq \| (u-v_1) \partial_x w_1 \|_{H^{s-1}} + \| (u-v_1) \partial_x w_2 \|_{H^{s-2}}\\ &\leq c \| u-v_1 \|_{H^{s-1}} \left( \| \partial _x w_1\|_{H^{s-1}} + \| \partial _x w_2\|_{H^{s-2}} \right)\\ &\leq c \| U- V \|_{H^{s-1} \times H^{s-2}} \| W \|_{H^{s-1} \times H^{s-2}}. \end{align*} Taking $V=0$ in \eqref{e3.5}, we deduce that $A(U) \in L\left( H^s\times H^{s-1}, H^{s-1}\times H^{s-2} \right)$. \end{proof} \begin{lemma}[\cite{EsLeYi}] \label{lem3.7} Let $B(U) = Q A(U) Q^{-1} - A(U)$, for $U \in H^s \times H^{s-1}$ $(s\geq 2)$. Then $B(U) \in L \left( H^{s-1} \times H^{s-2} \right)$ and \begin{gather*} \|\left( B(U) - B(V) \right) W \|_{H^{s-1} \times H^{s-2}} \leq \mu_2 \|U-V\|_{H^s \times H^{s-1}} \|W\|_{H^{s-1} \times H^{s-2}}, \\ U, V\in {H^s \times H^{s-1}},\ W \in {H^{s-1} \times H^{s-2}}. \end{gather*} \end{lemma} \begin{lemma}[\cite{EsLeYi}] \label{lem3.8} Let $U \in H^s \times H^{s-1}\ (s\geq 2)$. Then the operator $f(U)$ defined by \eqref{e3.4} is bounded on bounded sets in $(H^{s-1} \times H^{s-2})$, and satisfies \begin{itemize} \item[ (a)] $ \| f(U) - f(V) \|_{H^{s} \times H^{s-1}} \leq \mu_3 \|U-V\|_{H^s \times H^{s-1}}$, $U, V \in {H^s \times H^{s-1}}$, \item[(b)] $\| f(U) - f(V) \|_{H^{s-1} \times H^{s-2}} \leq \mu_4 \|U-V\|_{H^{s-1} \times H^{s-2}}$, $U, V\in {H^s \times H^{s-1}}$. \end{itemize} \end{lemma} \begin{proof}[Proof of Theorem \ref{thm3.2}] The result follows from Lemmas \ref{lem3.5}--\ref{lem3.8}. \end{proof} \section{Blow-up mechanism for $b=2$} In this section, we investigate the problem of the wave-breaking phenomenon for the initial-value problem of the periodic two-component Dullin-Gottwald-Holm system which is a special case of \eqref{e1.5} as $b=2$. \subsection{Preliminaries} The periodic two-component Dullin-Gottwald-Holm system can be written as \begin{equation}\label{e4.1} \begin{gathered} u_t-u_{txx}-Au_x+\gamma u_{xxx}+3u u_x-2u_x u_{xx}-u u_{xxx} +\rho \rho_x=0, \quad t>0,\; x\in \mathbb{R},\\ \rho_t+(u\rho)_x=0, \quad t>0,\; x\in \mathbb{R}, \\ u(0,x)=u_0(x), \quad x \in\mathbb{R},\\ \rho(0,x)=\rho_0(x), \quad x\in \mathbb{R},\\ u(t,x+1)=u(t,x), \quad t\ge0,\; x\in \mathbb{R},\\ \rho(t,x+1)=\rho(t,x), \quad t\ge0,\; x\in \mathbb{R}. \end{gathered} \end{equation} Let $G(x)= \frac{\cosh(x-[x]-1/2)}{2\sinh(1/2)}, x \in \mathbb{S}$. Then $(1-\partial^2_x)^{-1}f=G*f$ for all $f\in L^2(\mathbb{S})$, $u=G*m$ and $m=u-u_{xx}$. Our system \eqref{e4.1} can be written in the following ``transport'' type \begin{equation} \label{e4.2} \begin{gathered} u_t + (u- \gamma) u_x = -\partial_x G \ast \Big( u^2 + {1\over2}u^2_x + (\gamma -A) u + \frac 12 \rho^2 \Big), \quad t>0,\quad x\in \mathbb{R},\\ \rho_t+(u\rho)_x=0, \quad t>0,\; x\in \mathbb{R}, \\ u(0,x)=u_0(x), \quad x \in\mathbb{R},\\ \rho(0,x)=\rho_0(x), \quad x\in \mathbb{R},\\ u(t,x+1)=u(t,x), \quad t\ge0,\; x\in \mathbb{R},\\ \rho(t,x+1)=\rho(t,x), \quad t\ge0,\; x\in \mathbb{R}. \end{gathered} \end{equation} To study the wave-breaking problem, we now briefly give the needed results without proof to pursue our goal. We consider the following two associated Lagrangian scales of the system \eqref{e4.1} \begin{equation} \label{e4.3} \begin{gathered} \frac{\partial {q_1}}{\partial t}=u(t,q_1)-\gamma, \quad 00,\quad (t,x)\in [0,T)\times {\mathbb{R}},\\ q_{2x}(t,x)=\exp \Big(\int^t_0 u_x(\tau,q_2(\tau,x))d \tau\Big)>0,\quad (t,x)\in [0,T)\times {\mathbb{R}}. \end{gather*} \end{lemma} The above lemmas indicate that $q_1(t,\cdot):\mathbb{R}\to \mathbb{R} $ and $q_2(t,\cdot):\mathbb{R}\to \mathbb{R} $ are diffeomorphisms of the line for each $t\in[0,T)$. Hence, the $L^\infty$ norm of any function $v(t,\cdot)\in L^\infty(\mathbb{S})$ is preserved under the family of diffeomorphisms $q_1(t,\cdot)$ and $q_2(t,\cdot)$ with $t\in[0,T)$; that is, \begin{equation} \label{e4.5} \|v(t,\cdot)\|_{L^\infty(\mathbb{S})}=\|v(t,q_1(t,\cdot))\|_{L^\infty(\mathbb{S})} =\|v(t,q_2(t,\cdot))\|_{L^\infty(\mathbb{S})},\quad t\in[0,T). \end{equation} Similarly, we have \begin{gather} \label{e4.6} \inf_{x\in \mathbb{S}} v(t,x)=\inf_{x\in \mathbb{S}} v(t,q_1(t,x)) =\inf_{x\in \mathbb{S}} v(t,q_2(t,x)),\quad t\in[0,T), \\ \label{e4.7} \sup_{x\in \mathbb{S}} v(t,x)=\sup_{x\in \mathbb{S}} v(t,q_1(t,x)) =\sup_{x\in \mathbb{S}} v(t,q_2(t,x)),\quad t\in[0,T). \end{gather} \begin{lemma}[\cite{EsLeYi}] \label{lem4.2} Let $(u,\rho)$ be the solution of \eqref{e4.1} with initial data $(u_0,\rho_0) \in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S}), s\geq2$, and $T$ the maximal time of existence. Then we have \begin{equation} \label{e4.8} \rho(t,q_2(t,x))q_{2x}(t,x)=\rho_0(x),\quad (t,x)\in[0,T)\times \mathbb{S}. \end{equation} Moreover if there exists $x_0\in \mathbb{S}$ such that $\rho_0(x_0)=0$, then $\rho(t,q_2(t,x_0))=0$ for all $t\in[0,T)$. \end{lemma} \begin{lemma}[\cite{CoEs1}] \label{lem4.3} Let $T>0$ and $v\in C^1([0,T);H^2(\mathbb{R}))$. Then for every $t\in[0,T)$, there exists at least one point $\xi(t)\in{\mathbb{R}}$ with $$ m(t):= \inf_{x\in \mathbb{R}}\left(v_x(t,x)\right)=v_x(t,\xi(t)). $$ The function $m(t)$ is absolutely continuous on $(0,T)$ with $$ \frac{dm(t)}{dt}=v_{tx}(t,\xi(t))\quad a.e.\quad on \quad (0,T). $$ \end{lemma} We may use the following lemma derived in \cite{GuLi1} to establish the blow-up criterion of solution to \eqref{e4.1}. \begin{lemma} \label{lem4.4} Let $00$. Define $$ t_1=\max\{t0$ such that $$ u_x(t,x)\ge -M , \quad \forall (t,x)\in [0,T)\times \mathbb{S}. $$ It is now inferred from Lemma \ref{lem4.12} that $|u_x(t,x)|\le C$, where $$ C=C(A,\gamma,M,\|(u_0,\rho_0)\|^2_{H^s\times H^{s-1}}). $$ Therefore, Theorem \ref{thm4.5} in turn implies that the maximal existence time $T=\infty$, which contradicts the assumption that $T<\infty$. Conversely, the Sobolev embedding theorem $H^s(\mathbb{S})\hookrightarrow L^{\infty}(\mathbb{S})$ with $s>1/2$ implies that if \eqref{e4.10} holds, the corresponding solution blows up in finite time. This completes the proof. \end{proof} Now, we give the following theorems with some initial conditions which guarantee wave breaking in finite time. \begin{theorem} \label{thm4.14} Let $(u,\rho)$ be the solution of \eqref{e4.1} with the initial data $(u_0,\rho_0) \in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S}),s\geq2$, and $T$ the maximal time of existence. Assume that there is some $x_0\in \mathbb{S}$ such that $$ \rho_0(x_0)=0,\quad u_{0,x}(x_0)=\inf_{ x\in\mathbb{S}} u_{0,x}(x), $$ and \begin{equation} \label{e4.29} u_{0,x}(x_0)<-C_1, \end{equation} where $C_1$ is defined as $$ C^2_1=\Big((1-\kappa){\frac{e+1}{e-1}}+{1\over2}\Big)\|(u_0,\rho_0) \|^2_{H^1\times L^2}+\frac{(-1+\sinh 1)(\gamma-A)^2}{4 \sinh ^2(1/2)}. $$ Then the corresponding solution to system \eqref{e4.1} blows up in the following sense: there exists a $T_1$ with \begin{equation} \label{e4.30} 05/2$. There exists a $T^*$ with $00$, \item[(b)] $ \liminf_{t\to T^{*}} \{\inf_{x\in\mathbb{S}} \rho_x(t,x)\} =-\infty$, if $\rho_{0,x}(x_0)<0$. \end{itemize} \end{corollary} \begin{proof} With the assumptions of Theorem \ref{thm4.14}, we have $$ \rho_0(x_0)=0,\quad u_{0,x}(x_0)=\inf_{ x\in\mathbb{S}} u_{0,x}(x), $$ and $u_{0,x}(x_0)<-C_1$. Evaluating $\rho$ along the trajectory $q_2(t,x)$, we obtain $$ {{d\rho_x\left(t,q_2(t,x)\right)}\over{dt}}=-u_{xx} \left(t,q_2(t,x)\right){\rho\left(t,q_2(t,x)\right)} -2u_{x}\left(t,q_2(t,x)\right){\rho_x\left(t,q_2(t,x)\right)}. $$ As in the proof of Theorem \ref{thm4.14}, we can choose $x_2(t)\in \mathbb{S}$ such that $q_2(t,x_2(t))=\xi(t),t\in [0,T)$. Then we have $$ m(t):=u_x(t,\xi(t))=\inf_{x \in \mathbb{S}}(u_x(t,x)),\quad t\in [0,T). $$ Hence, $u_{xx}(t,\xi(t))=0, \;a.e.\quad t\in [0,T)$. This in turn implies $$ {{d\rho_x\left(t,\xi(t)\right)}\over{dt}} =-2u_{x}\left(t,\xi(t)\right){\rho_x\left(t,\xi(t)\right)}, $$ and $$ \rho_x(t,\xi(t))=\rho_{0,x}(x_0)e^{ -2\int^t_0{u_x(\tau,\xi(\tau))d\tau}} =\rho_{0,x}(x_0)e^{ -2\int^t_0{{\underset{x\in\mathbb{S}}{\inf}u_x(\tau,x)d\tau}}}. $$ Since $m(t)$ is strictly decreasing in $[0,T)$, by \eqref{e4.36} we have $$ e^{ -2\int^t_0{{\underset{x\in\mathbb{S}}{\inf}u_x(\tau,x)d\tau}}} \geq{e^{ -2\int^t_0{{\frac {u_{0,x}(x_0)}{1+\delta u_{0,x}(x_0)\tau}d\tau}}}}\geq {e^{{-\frac{2}{\delta}}\ln\left(1+\delta u_{0,x}(x_0)t\right)}}, $$ where $\delta$ is defined in \eqref{e4.34}. So $$ {e^{{-\frac{2}{\delta}}\ln(1+\delta u_{0,x}(x_0)t)}}\to +\infty, $$ if $t\to{-\frac {1}{\delta u_{0,x}(x_0)}}$. Therefore, it is inferred from \eqref{e4.37} that there exists some $T^*$ with $00$ ($C_0=\|(u_0,\rho_0)\|^2_{H^1\times L^2}$) such that \begin{equation} \label{e4.38} \int_\mathbb{S}{u_{0x}^3}dx<-K_0, \end{equation} then the corresponding solution to \eqref{e4.1} blows up in finite time. \end{theorem} \begin{proof} Applying $u_x^2\partial_x$ to both sides of the first equation in \eqref{e4.2} and integrating by parts with the fact that $$ -3\int_\mathbb{S}{uu_x^2u_{xx}}dx=\int_\mathbb{S}{u_x^4}dx. $$ We have \begin{equation} \label{e4.39} \begin{aligned} {{d\over{dt}}\int_\mathbb{S}{u_x^3}dx}+{1\over2}\int_\mathbb{S}{u_x^4}dx &=3\int_\mathbb{S}{u_x^2\Big(u^2+(\gamma-A)u+{{1\over2}\rho^2}\Big)}dx\\ &\quad -3\int_\mathbb{S}{{u_x^2}G* \Big(u^2+{1\over2}u_x^2+(\gamma-A)u+{{1\over2}\rho^2}\Big)}dx. \end{aligned} \end{equation} Note that $$ \big|\int_\mathbb{S}{u_x^3}dx \big| \leq \Big(\int_\mathbb{S}{u_x^4}dx \Big)^{1/2} \Big(\int_\mathbb{S}{u_x^2}dx \Big)^{1/2}, $$ and $ C_0=\|(u_0,\rho_0)\|^2_{H^1\times L^2}$. Thus we have \begin{equation} \label{e4.40} \int_\mathbb{S}{u_x^4}dx\geq{1\over{C_0}}{\Big(\int_\mathbb{S}{u_x^3}dx \Big)^2}. \end{equation} Using Corollary \ref{coro4.9}, we obtain the estimate \begin{equation} \label{e4.41} \int_\mathbb{S}{u_x^2 u^2} dx \leq {\|{u}\|}^2_{L^{\infty}(\mathbb{S})}{ \int_\mathbb{S}{u_x^2}dx } \leq{e+1\over{2(e-1)}}C_0^2. \end{equation} By the assumption $\int_\mathbb{S}{\rho_0(x)dx=0}$ and Lemma \ref{lem4.2}, we have $$ \int_\mathbb{S}{\rho(t,x)dx}=\int_\mathbb{S}{\rho_0(x)dx=0}. $$ It then follows that for any $t\in[0,T)$, there exists $x_3(t)\in\mathbb{S}$ and $\rho(t,x_3(t))=0$. It is noted that $$ \rho(t,x)=\int^{x(t)}_{x_3(t)}{\rho_x (t,s)} ds, \quad x_3(t), x(t)\in\mathbb{S}, $$ which implies that \begin{gather} |\rho(t,x)|\leq \big|\int^{x(t)}_{x_3(t)}{\rho_x (t,s)} ds \big|\leq M , \nonumber\\ \label{e4.42} \int_\mathbb{S}{u_x^2\rho^2}dx \leq M^2{\int_\mathbb{S}{u_x^2}dx}\leq M^2C_0, \\ \label{e4.43} \big|\int_\mathbb{S}{u_x^2 u}dx \big| \leq{\|{u}\|}_{L^{\infty}(\mathbb{S})}{ \int_\mathbb{S}{u_x^2}dx } \leq \Big({e+1\over{2(e-1)}} \Big)^{1/2}C_0^{3/2}, \end{gather} and \begin{equation} \label{e4.44} \begin{aligned} &\int_\mathbb{S}{{u_x^2}G*(\gamma-A)u}dx\\ &\geq-|\gamma-A|\|G\|_{L^{\infty}(\mathbb{S})} \|u\|_{L^{\infty}(\mathbb{S})}\int_\mathbb{S}{u_x^2}dx\\ &\geq-|\gamma-A|{\frac{\cosh(1/2)}{2\sinh(1/2)}} \Big({e+1\over{2(e-1)}} \Big)^{1/2}C_0^{3/2} =-|\gamma-A|\Big({e+1\over{2(e -1)}}\Big)^{3/2}C_0^{3/2}. \end{aligned} \end{equation} In view of the above inequality \eqref{e4.41}, \eqref{e4.42}, \eqref{e4.43} and \eqref{e4.44}, it follows from Lemma \ref{lem4.10} that \begin{equation} \label{e4.45} \begin{aligned} {{d\over{dt}}\int_\mathbb{S}{u_x^3}dx} &\leq -{1\over2{C_0}}{ \Big(\int_\mathbb{S}{u_x^3}dx \Big)^2} +3\int_\mathbb{S}{u_x^2 \Big(u^2+{{1\over2}\rho^2} \Big)}dx +3(\gamma-A)\int_\mathbb{S}{u_x^2 u}dx \\ &\quad -3\int_\mathbb{S}{{u_x^2}G* \Big(u^2+{1\over2}u_x^2 \Big)} +{{u_x^2}G*(\gamma-A)u}+{{u_x^2}G* \left ({1\over2}\rho^2 \right )}dx\\ \leq & -{1\over2{C_0}}{ \Big(\int_\mathbb{S}{u_x^3}dx \Big)^2} +3\int_\mathbb{S}{u_x^2 \left (u^2+{{1\over2}\rho^2} \right )}dx+3(\gamma-A)\int_\mathbb{S}{u_x^2 u}dx\\ &\quad -3\kappa\int_\mathbb{S}{u_x^2u^2}dx-3\int_\mathbb{S}{{u_x^2}G*(\gamma-A)u}dx\\ &= -{1\over2{C_0}}{\left (\int_\mathbb{S}{u_x^3}dx \right )^2} +3(1-\kappa)\int_\mathbb{S}{u_x^2u^2}dx+\frac{3}{2}\int_\mathbb{S}{u_x^2{\rho^2}}dx\\ &\quad +3|\gamma-A|\int_\mathbb{S}{u_x^2 u}dx-3\int_\mathbb{S}{{u_x^2}G*(\gamma-A)u}dx\\ &\leq -{1\over2{C_0}}{\Big(\int_\mathbb{S}{u_x^3}dx \Big)^2} +{{3(1-\kappa)(e+1)}\over{2(e-1)}}C_0^2+{\frac{3}{2}}M^2C_0\\ &\quad +{{3(3e-1)}\over{2(e-1)}}|\gamma-A|\Big({e+1\over{2(e-1)}}\Big)^{1/2}C_0^{3/2}. \end{aligned} \end{equation} Set $ h(t)=\int_\mathbb{S}{u_x^3}dx $, and \begin{align*} K^2={{3(1-\kappa)(e+1)}\over{2(e-1)}}C_0^2+{\frac{3}{2}}M^2C_0 +{{3(3e-1)}\over{2(e-1)}}|\gamma-A| \left({e+1\over{2(e-1)}}\right)^{1/2}C_0^{3/2}. \end{align*} Note that if $h(0)<-{\sqrt{2C_0}}K$, then $h(t)<-{\sqrt{2C_0}}K$. Therefore, we can solve the above inequality \eqref{e4.45} to obtain $$ {{h(0)+{\sqrt{2C_0}}K}\over{h(0)-{\sqrt{2C_0}}K}} e^{\sqrt{2\over{C_0}}Kt}-1\leq{{{2\sqrt{2C_0}}K}\over{h(0)-{\sqrt{2C_0}}K}}\leq0. $$ Due to the inequality $$ 0<{{h(0)+{\sqrt{2C_0}}K}\over{h(0)-{\sqrt{2C_0}}K}}<1, $$ then there exists $T_1$ satisfying $$ 0-M$. Let $K_0=\sqrt{2C_0}K$. As a result, we deduce that the solution blows up in finite time which is the desired result in the theorem. \end{proof} Next, we give a wave breaking result when the initial profile $u_0$ is odd and $\rho_0$ is even. \begin{theorem} \label{thm4.17} Let $(u,\rho)$ be the solution of \eqref{e4.1} with the initial data $(u_0,\rho_0) \in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S}),s\geq2$, and $T$ the maximal time of existence. Assume that $u_0\not\equiv 0$ is odd, $\rho_0$ is even, $u_{0,x}\leq0$ and $\rho_0(0)=0$. Assume $\gamma=A=0$. Then the corresponding solution to system \eqref{e4.1} blows up in finite time. \end{theorem} \begin{proof} Similar to the proof of Lemma \ref{lem4.12}, it suffices to consider $s\ge 3$. Since $u_0$ is odd and $\rho_0$ is even, the corresponding solution $(u(t,x),\rho(t,x))$ satisfies that $u(t,x)$ is odd and $\rho(t,x)$ is even with respect to $x$ for given $00$ is the lifespan of the corresponding solution to \eqref{e4.1}. Assume further there is some $x_0\in S$ such that $$ \rho_0(x_0)=0,\quad u_{0,x}(x_0)=\underset{ x\in\mathbb{S}}\inf u_{0,x}(x). $$ If $T_{\rm max}<\infty$, then the lifespan $T_{\rm max}>0$ satisfies \begin{equation} \label{e4.52} T_{\rm max}\geq\overline{T}=\sqrt{2\over{C_2}}\arctan \Big({-\sqrt{2C_2}}\over{\inf_{x\in\mathbb{S}} u_{0,x}(x)}\Big) \end{equation} where $$ C_2={{5e+3}\over{4(e-1)}}\|(u_0,\rho_0)\|^2_{H^1\times L^2}+\frac{(-1+\sinh 1)(\gamma-A)^2}{8 \sinh ^2(1/2)} $$ is defined in\eqref{e4.14}. \end{theorem} \begin{proof} Let us first assume that the initial data $(u_0,\rho_0) \in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S}),s\geq3$. In view of \eqref{e4.33}, we have $$ m'(t)=-\frac{1}{2}m^2(t)+f(t,q_2(t,x_2))\geq-\frac{1}{2}m^2(t)-C_2. $$ Integrating this inequality, we obtain $$ {\arctan{m(t)\over{\sqrt{2C_2}}}}\geq \arctan{m(0)\over{\sqrt{2C_2}}}-{\sqrt{C_2\over2}}t,\quad\quad \forall t<\min(T_{\rm max},\overline{T}). $$ This in turn implies that $$ m(t)\geq{{\sqrt{2C_2}m(0)-2C_2\tan\big({\sqrt{C_2\over2}}t\big)} \over{{\sqrt{2C_2}+m(0)\tan({\sqrt{C_2\over2}}t)}}}. $$ Due to \eqref{e4.10}, there appears the result \eqref{e4.52} from the above inequality. If $s\in[2,3)$, it is easy to see the lifespan $T^s_{\rm max}$as a function of $s$ for the initial data $u_{0,x}(x_0)=\inf_{ x\in\mathbb{S}} u_{0,x}(x)$ with $s\geq2$ is nonincreasing. So $T^s_{\rm max}\geq T^r_{\rm max}$ for $2\leq s\leq r$. This ensures the validity of lower bound of the lifespan $T^s_{\rm max}$ in \eqref{e4.52} for all $s\geq 2$. \end{proof} \section{Existence of global solution} In this section, we provide a sufficient condition for the existence of a global solution of system \eqref{e4.1}. \begin{theorem} \label{thm5.1} Assume the initial data $(u_0, \rho_0-1 ) \in H^s \times H^{s-1}$, $s\geq2$. If \begin{equation} \label{e5.1} \inf_{x\in \mathbb{S}} \rho_0 (x) > 0, \end{equation} then the corresponding solution $(u,\rho)$ to the initial-value problem of system \eqref{e4.1}, as given by Theorem \ref{thm3.2}, exists globally in time. \end{theorem} \begin{proof} As before we prove this theorem for $s\geq 3$. By Theorem \ref{thm4.5}, to obtain global existence, it suffices to control $|u_x (t,x) |$. We will achieve this by proving the following key results. \begin{equation} \big| \inf_{x\in (\mathbb{S})} u_x(t,x) \big|,\quad \big| \sup_{x\in (\mathbb{S})} u_x(t,x) \big| \leq C_4 e^{C_3t},\label{e5.2} \end{equation} where \begin{gather*} C_3 =1+{\frac{5e+3}{4(e-1)}}\|(u_0,\rho_0)\|^2_{H^1\times L^2}+\frac{(-1+\sinh 1)(\gamma-A)^2}{8 \sinh ^2(1/2)}, \\ C_4 = \frac 1 {\inf_{x\in (\mathbb{S}} \rho_0 (x)} \left( 1 + \|u_{0,x} \|^2_{L^{\infty}} + \|\rho_0 \|^2_{L^{\infty}} \right). \end{gather*} We first estimate $| \inf_{x\in (\mathbb{S})} u_x(t,x)|$. Recall that $m(t)$, $\xi(t)$ and $x_2 (t)$ are defined by \begin{gather*} m(t):=u_x(t,\xi(t))=\inf_{x \in \mathbb{S}}(u_x(t,x)),\quad t\in [0,T),\\ u_{xx}(t,\xi(t))=0, \quad \text{a.e. } t\in [0,T). \end{gather*} We can choose $x_2(t)\in \mathbb{R}$, such that $q_2(t,x_2(t))=\xi(t)$. Let $\zeta (t) = \rho \big( t, \xi(t) \big)$. Evaluating \eqref{e4.16} along the trajectory $q_2 (t,x)$ at $\xi(t)$ leads to \begin{gather} m'(t) = -\frac12 m^2(t) + \frac12 \zeta^2 (t) + f(t,\xi(t)),\label{e5.3}\\ \zeta'(t) = -\zeta m,\quad t\in [0,T).\label{e5.4} \end{gather} In view of \eqref{e5.1}, it follows \eqref{e5.4} that $\zeta (t) $ and $\zeta(0)$ are all positive. We define the following Lyapunov function, which is due to Constantin and Ivanov \cite{CoIv} \begin{equation} w(t) = \zeta(t) + \frac 1 {\zeta(t)} ( 1 + m^2(t)). \label{e5.5} \end{equation} It is always positive in $[0,T)$ since $\zeta(t)$ and $\zeta(0)$ are all positive. Differentiating and using \eqref{e5.3} and \eqref{e5.4}, we obtain \begin{equation} \begin{aligned} w'(t) & = \zeta'(t) - \frac 1 {\zeta^2 (t)}(1+m^2(t)) \zeta '(t) + \frac 2{\zeta(t)} m'(t)m(t)\\ &= \frac {2 m(t)} {\zeta(t)} \left ( \frac12 + f(t,\xi(t)) \right ) \\ & \leq \frac 1 {\zeta(t)} (1+ m^2(t)) \left ( \frac12 + |f(t,\xi(t)))| \right )\\ &\leq C_3 w(t). \end{aligned} \label{e5.6} \end{equation} Solving \eqref{e5.6} and recalling the definitions of $C_3$ and $C_4$, we infer that \begin{equation} \begin{aligned} w (t) & \leq w(0) e^{C_3 t} = \frac 1 {\zeta(0)} \left(\zeta^2(0) +1 + m^2(0) \right) e^{C_3 t}\\ &\leq \frac 1 {\zeta(0)} \left( 1 + \|u_{0,x} \|^2_{L^{\infty} } + \|\rho_0\|^2_{L^{\infty}} \right) e^{C_3 t}\\ & = C_4 e^{C_3 t}. \end{aligned} \label{e5.7} \end{equation} It is easy to see that $\zeta(t) \leq w(t)$ and $| m(t)| \leq w (t)$. Therefore, for $t\in [0,T)$, \[ \big| \inf_{x \in(\mathbb{S})} u_x (t,x) \big| = |m(t)| \leq w(t) \leq C_4 e^{C_3 t}. \] To estimate $| \sup_{x\in (\mathbb{S})} u_x(t,x)|$, recalling $\bar m(t)$, $\eta(t) $ and $x_1(t)$ as defined in Lemma \ref{lem4.12}, let $\bar \zeta(t) = \rho (t,\eta(t))$. For $t\in [0,T)$, we obtain \begin{gather*} \bar m'(t) = -\frac12 \bar m^2(t) + \frac12 \bar\zeta^2 (t) + f(t,\eta(t)),\\ \bar \zeta'(t) = -\bar \zeta \bar m. \end{gather*} Define \[ \bar w(t) = \bar \zeta(t) + \frac 1 {\bar \zeta(t)} ( 1 +\bar m^2(t)). \] Similar to \eqref{e5.6} and \eqref{e5.7}, we have \[ \bar w(t) \leq C_3 \bar w(t) \quad \text{and} \quad \bar w(t) \leq C_4 e^{C_3 t}. \] Therefore, \[ \big| \sup_{x \in (\mathbb{S})} u_x (t,x) \big| = | \bar m(t)| \leq {\bar w(t)} \leq C_4 e^{C_3 t},\quad t\in [0,T). \] Therefore, the proof is complete. \end{proof} \subsection*{Acknowledgements} M. Zhu was partially supported by the JSPS Innovation Program, under the Grant CXZZ12-0084. J. 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