\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 224, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/224\hfil Approximate solutions to BBM equations] {Approximate solutions to BBM equations with bilinear control in a slowly varying medium} \author[W. Chen, L. Tian, G. Xu, P. Yang \hfil EJDE-2017/224\hfilneg] {Wenxia Chen, Lixin Tian, Gang Xu, Ping Yang} \address{Wenxia Chen \newline Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu, 212013, China} \email{chenwx@ujs.edu.cn} \address{Lixin Tian \newline Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu, 212013, China. \newline School of Mathematical Science, Nanjing Normal University, Nanjing, Jiangsu, 210023, China} \email{tianlx@ujs.edu.cn, tianlixin@njnu.edu.cn} \address{Gang Xu (corresponding author) \newline Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu, 212013, China} \email{gxu@ujs.edu.cn} \address{Ping Yang \newline Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu, 212013, China} \email{598761315@qq.com} \dedicatory{Communicated by Zhaosheng Feng} \thanks{Submitted December 11, 2016. Published September 19, 2017.} \subjclass[2010]{35Q53, 35B40} \keywords{Approximate solution; BBM equation; bilinear control; \hfill\break\indent slowly varying medium} \begin{abstract} This article concerns an approximate solution for the Benjamin-Bona-Mahony (BBM) equation with a bilinear control in slowly varying medium. By a sharp estimation of the error term, a suitable approximate solution for this equation is established. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this article, we consider the following BBM equation with internal bilinear control in slowly varying medium \begin{equation} \label{e1.1} ( {1 - \lambda \partial _x^2} ){\partial _t}u + {\partial _x} ( {\partial _x^2u - u + {m_\varepsilon }{u^2}} ) = f(t,x),\quad (t,x) \in{\mathbb{R \times R}} \end{equation} where $u=u(t,x)$ is a real-valued function, $\lambda \in (0,1)$ is a constant, and the interior control $f$ is given by the bilinear control (or feedback law) $f(t,x)=n(t,x)u(t,x)$ with $n(t,\cdot )\in {{C}^{3}}(\mathbb{R})\cap {{L}^{2}}(\mathbb{R})\cap {{L}^{\infty }} (\mathbb{R})$. Concerning slowly varying medium ${m_\varepsilon } = m( {\varepsilon x} )$, we always assume that there exist positive constants $k$ and $\gamma$ such that \begin{equation} \label{e1.2} \begin{gathered} 1 < m(s) < 2,\quad m'(s) > 0,\quad \forall s \in {\mathbb{R}}\\ 0 < m(s) - 1 < k{e^{\gamma s}},\quad \forall s \le 0 \\ 0 < 2 - m(s) < k{e^{ - \gamma s}},\quad \forall s > 0. \end{gathered} \end{equation} Clearly, it is inferred from \eqref{e1.2} that $\lim _{s \to - \infty } m(s) = 1$ and $\lim _{s \to + \infty } m(s) = 2$. Let us briefly review some results concerning the related control problems and stability of solitons for the BBM, KdV and gKdV equations. The BBM equation model \cite{s1} was proposed by Benjamin, Bona and Mahony. Ko and Kuehl \cite{s2} considered the approximate solution of the soliton for the variable coefficient KdV equation under given initial conditions, and obtained the unavoidable loss of the solitary wave energy in the propagation process. Bona, Pritchard and Scott \cite{s3} introduced solitary wave interaction in the dispersion medium numerically. Albert \cite{s4,s5} investigated the global existence, the long time behavior and the decay of solutions about the gBBM equation. Weinstein \cite{s6} studied the existence and dynamics stability of solitary wave. Weinstein \cite{s7} also obtained asymptotic stability of regularized long wave equations. Russell and Zhang \cite{s8,s9} studied controllability and stabilizability of the third-order linear dispersion equation and KdV equation, and showed smoothing and decay properties of the KdV equation on a periodic domain \cite{s10}. The asymptotic stability of the soliton solution of the BBM equation in ${H^1}$ space is studied in \cite{s11}. Dejak and Jonsson \cite{s12} considered the long-time dynamics of the variable coefficient modified KdV solitary waves. In the same year, Dejak and Sigal \cite{s13} studied the KdV solitary waves over a variable bottom similarly. Martel \cite{s14} researched asymptotic N-solitons-like solutions of the subcritical and critical gKdV equations. Martel and Merle has been making a great contribution for the BBM, KdV and gKdV equations. They proved that the soliton solution near ${Q_c}$ of gKdV equation with a general nonlinearity is asymptotically stable in ${H^1}$ space \cite{s15}. They also studied the inelastic interaction of nearly equal solitons for the BBM and gKdV equations in \cite{s16,s17}. They estimated the error term between the approximate solution and the exact solution. They also gave descriptions of the inelastic collision of two solitary waves for the BBM and quartic gKdV equations \cite{s18,s19}, and estimated some non-zero residual items accurately. $ Mu\widetilde{n}oz$ explored soliton dynamics and the existence and global properties under slowly varying medium for the gKdV equation, and proved that there is no pure soliton solution \cite{s20}. On this basis, he introduced inelastic character of solitons of slowly varying gKdV equations in \cite{s21}, and dealt with approximate controllability of the gKdV solitons with bilinear control in \cite{s22, s23}. Holmer introduced dynamics of the KdV solitons in the presence of a slowly varying potential, and obtained an explicit description of the trajectory of the soliton parameters of scale and position on the dynamically relevant time scale, together with an estimate on the error \cite{s24}. Unlike the previous study, in this paper, we focus on a bilinear control problem for a given BBM soliton in slowly varying medium. The main difficulties to our problem are the suitable construction and the decomposition of an approximate solution. An essential step of the proof is the error term between the approximate solution and exact solution can be controlled under $O( {{\varepsilon ^{3/2}}{e^{ - \gamma \varepsilon | {\rho ( t )} |}}} )$ during an interval of time $[ {0,T} ]$. Finally, we introduce a cut-off function $\zeta \in {C^\infty }( {\mathbb{R}} )$ and redefine the new approximate solution $\widetilde K= {\zeta _\varepsilon }( y )K( {t,x} )$ to solve the problem of $K \notin {L^2}( {\mathbb{R}} )$. The rest of this paper is organized as follows. In Section 2, we introduce the soliton solution of the BBM equation and the associated control system. In Section 3, we construct an approximate solution and prove that the error term can be raised to $O(\varepsilon^{2})$ by analyzing the first order term. In Section 4, we introduce a cut-off function $\zeta \in {C^\infty }( {\mathbb{R}} )$ in order to resolve the case of $K \notin {L^2}( {\mathbb{R}} )$. \section{Preliminaries} We consider the equation \begin{equation} \label{e2.1} ( {1 - \lambda \partial _x^2} ){u_t} + {( {{u_{xx}} - u + {m_\varepsilon }{u^2}} )_x} = nu, \end{equation} which has solitary wave solutions: \begin{equation} \label{e2.2} u( {t,x} ) = {Q_c}( {x - ct} ), \end{equation} and \begin{equation} \label{e2.3} {Q_c}( x ) = ( {1 + c} )Q\Big( {\sqrt {\frac{{1 + c}}{{1 + \lambda c}}} x} \Big), \end{equation} where the parameter $ c > 0 $ describes the wave speed of the soliton. Differentiating \eqref{e2.3} with respect to $x$ gives \begin{equation} \label{e2.4} \big(Q_c(x)\big) _{x}'= \frac{{{{( {1 + c} )}^{3/2}}}}{{{{( {1 + \lambda c} )} ^{1/2}}}}Q'(\xi),\quad \xi=\sqrt{\frac{1+c}{1+\lambda c}}x. \end{equation} Differentiating \eqref{e2.3} with respect to $c$ leads to \begin{equation} \label{e2.5} \wedge {Q_c}(x) = \big(Q_c(x)\big) _{c}'= \frac{1}{{1 + c}} \Big(Q_{c}(x)+\frac{1-\lambda}{2(1+\lambda c)}x \big(Q_c(x)\big) _{x}'\Big). \end{equation} By \eqref{e2.3}, we have \begin{equation} \label{e2.6} Q( x ) = \frac{1}{{1 + c}}{Q_c}\Big( {\sqrt {\frac{{1 + \lambda c}}{{1 + c}}} x} \Big), \end{equation} with \begin{equation} \label{e2.7} Q( x ) = \frac{3}{2}{\cosh ^{ - 2}}( {\frac{x}{2}} )\quad\text{which satisfies } {Q''} + {Q^2} = Q, \end{equation} and \begin{equation} \label{e2.8} ( {1 + \lambda c} )Q_c'' + Q_c^2 = ( {1 + c} ){Q_c}. \end{equation} We now introduce the control $n( {t,x} ) = - \varepsilon n_0'( x ){Q_{c( t )}}( {x - \rho ( t )} )$. For any $ \varepsilon > 0$ small enough, we define \begin{gather} \label{e2.9} \alpha : = - \frac{4}{3}\frac{{\int_R {{Q^3}} }}{{\int_R {{Q^2}} }}> 0, \\ \label{e2.10} {n_\infty }: = - \frac{1}{\alpha }\log c. \end{gather} We choose a smooth function ${n_0}$ satisfying \begin{equation} \label{e2.11} \begin{gathered} {{n}_{0}}\in {{C}^{3}}({\mathbb{R}})\cap {{L}^{\infty }}({\mathbb{R}}), \\ | {{n}_{0}}(x) |\le k{{e}^{{{\gamma }_{0}}x}},\quad\text{for } x\le -1, \\ | {{n}_{\infty }}-{{n}_{0}}(x) |\le k{{e}^{-{{\gamma }_{0}}x}},\quad\text{for } x\ge 1, \\ | n_{0}^{(p)}(x) |\le k{{e}^{-{{\gamma }_{0}}| x |}},\quad x\in {\mathbb{R}},\; p=1,2,3, \\ {{{{n}'}}_{0}}(x)>0\quad\text{if }{{n}_{\infty }}>0;\quad {{{{n}'}}_{0}}(x)<0\quad \text{if } {{n}_{\infty }}<0. \end{gathered} \end{equation} for a fixed positive constant ${{\gamma }_{0}}$. Note that with this choice, it holds ${{\| {{n}_{0}} \|}_{\infty }}=| {{n}_{\infty }} |$. \section{Approximate solution} In the aforementioned context, we consider the function ${V_{c( t )}}(\varepsilon t,x) \in {L^\infty }( {\mathbb{R}} )$ satisfying the following hypothesis : \begin{itemize} \item[(H1)] $ V_{c( t )}'(\varepsilon t,x) \in {L^2}( {\mathbb{R}})$, ${\partial _c}{V_{c( t )}}(\varepsilon t,x) \in {L^\infty }( {\mathbb{R}} )$ \item[(H2)] For all $ t \in {\mathbb{R}}$, there exist positive constants $ k$ and $\gamma$ satisfying \[ {\| {{V_{c( t )}}(\varepsilon t,x)} \|_{{L^\infty }( {\mathbb{R}} )}} \le k{e^{ - \gamma \varepsilon | {\rho ( t )} |}}. \] \end{itemize} Equation \eqref{e1.1} remains invariant under space and time translations at the ${{H}^{1}}$-level, usually called mass conservation and energy conservation: \begin{equation} \label{e3.1} M( {u( t )} ) = \int_R {( {\frac{1}{2}{u^2} - \lambda u_x^2} )} ( {t,x} )dx = M( {u( 0 )} ), \end{equation} and \begin{equation} \label{e3.2} \begin{aligned} E(u(t))&=\frac{1}{2}\int_R {( {1 - \lambda \partial _x^2} )} \big( {{{( {{u_x}} )}^2} + {u^2}} \big)( {t,x} )dx - \frac{1}{3}\int_R {( {1 - \lambda \partial _x^2} )} {u^3}( {t,x} )dx\\ &=E(u(0)). \end{aligned} \end{equation} We introduce the time of interaction for any given $\varepsilon >0$ and ${{\delta }_{0}}>0$ small enough: \begin{equation} \label{e3.3} T:=\min \{ {{T}_{0}},{{\varepsilon }^{-1-{{\delta }_{0}}}} \}, \end{equation} where ${{T}_{0}}>0$ is the maximal time of existence for the solution $u(t)$. Let \begin{equation} \label{e3.4} y:=x-\rho(t),\quad R(t,x):=\frac{{{Q_{c( t )}}( y )}}{{m( {\varepsilon \rho ( t )} )}}, \end{equation} with $$ \rho ( t ) = \int_0^t {c( { \varepsilon s} )} ds,\quad \partial_{t}\rho(t)=c(\varepsilon t). $$ Define $K( {t,x} )$ as the approximate solution of equation \eqref{e2.1}: \begin{equation} \label{e3.5} K( {t,x} ): = R( {t,x} ) + W( {t,x} ),\quad W( {t,x} ): = \varepsilon n_0'( {\varepsilon \rho ( t )} ){V_{c( t )}}( {\varepsilon t,y} ), \end{equation} where ${V_{c( t )}}$ satisfies the hypothesis (H1)-(H2). Then we can reduce the error by introducing $K( {t,x} )$ defined in \eqref{e3.5}. Set \begin{equation} \label{e3.6} S[ K ]( {t,x} ): = ( {1 - \lambda \partial _x^2} ){K_t} + {( {{K_{xx}} - K + {m_\varepsilon }{K^2}} )_x}+ \varepsilon n_0'( {\varepsilon x} ){{{Q_{c( t )}} ( y )}}K, \end{equation} with ${m_\varepsilon } = m( {\varepsilon x} )$. \begin{theorem} \label{thm3.1} There exists a function ${{V_{c( t )}}} \in {L^\infty }( {\mathbb{R}} )$ such that $K( {t,x} )$ defined by \eqref{e3.5} satisfies \begin{equation} \label{e3.7} S[ K ]( {t,x} ) = ( {1 - \lambda \partial _x^2} ) \big( {{c'}{\partial _c}K - c{\partial _y}K} \big) + {S_0}[ K ] ( {t,x} ) \end{equation} for some $n \in {\mathbb{R}}$, and \begin{equation} \label{e3.8} {\| {{S_0}[ K ]( {t,x} )} \|_{{H^1}( {y > - \frac{2}{\varepsilon }} )}} \le {\varepsilon ^{3/2}}{e^{ - \gamma \varepsilon | {\rho ( t )} |}} + {\varepsilon ^3} \end{equation} and \begin{equation} \label{e3.9} \begin{aligned} &\big| {\int_{\mathbb{R}} {( {1 - \lambda \partial _x^2} ){{{Q_{c( t )}}( y )}}{S_0} [ K ]( {t,x} )dx} } \big| + \big| {\int_{\mathbb{R}} {( {1 - \lambda \partial _x^2} )y{{{Q_{c( t )}}( y )}} {S_0}[ K ]( {t,x} )dx} } \big| \\ &\le {\varepsilon ^2}{e^{ - \varepsilon \gamma | {\rho ( t )} |}} + {\varepsilon ^3}. \end{aligned} \end{equation} Following the strategy in \cite{s22}, from \eqref{e3.6} we obtain \begin{equation} \label{e3.10} S[K]= I+II +III, \end{equation} where $I= S[R](t,x)$, \[ II= ( {1 - \lambda \partial _x^2} ){W_t} + {( {{W_{xx}} - W + 2{m_\varepsilon }RW} )_x} + \varepsilon n_0'( {\varepsilon x} ){Q_{c( t )}}( y )W, \] and $III= {( {{m_\varepsilon }{W^2}} )_x}$, where \[ R(t,x)=\frac{{{Q_{c( t )}}( y )}}{{m( {\varepsilon \rho ( t )} )}}, \quad y=x-\rho(t). \] \end{theorem} For proving Theorem \ref{thm3.1}, we discuss $I$, $II$ and $III$, separately. \begin{lemma} \label{lem3.2} we have $$ I=\varepsilon {A_1}( {t,y} ) + {\varepsilon ^2}{A_2}( {t,y} ), $$ where \begin{equation} \label{e3.11} \begin{aligned} {A_1}( {t,y} ) &= \frac{{{c'}}}{{m( {\varepsilon \rho } )}}( {1 - \lambda \partial _x^2} ) \wedge {{{Q_{c( t )}}( y )}} - \frac{{{m'}( {\varepsilon \rho } )c}}{{{m^2}( {\varepsilon \rho } )}} ( {1 - \lambda \partial _x^2} ){{{Q_{c( t )}}( y )}}\\ &\quad + \frac{{{m'}( {\varepsilon \rho } )}}{{{m^2}( {\varepsilon \rho } )}} {\big( {y{Q^2_{c( t )}}( y )} \big)_x} + \frac{{n_0'( {\varepsilon \rho } )}}{{m( {\varepsilon \rho } )}}{Q^2_{c( t )}}( y ) \end{aligned} \end{equation} and \begin{equation} \label{e3.12} {A_2}( {t,y} ) = \frac{{{m''}( {\varepsilon \rho } )}}{{2{m^2} ( {\varepsilon \rho } )}}{\big( {{y^2}{Q^2_{c( t )}}( y )} \big)_x} + \frac{{n_0''( {\varepsilon \rho } )}}{{m( {\varepsilon \rho } )}}y{Q^2_{c( t )}}( y )~~~~~~~~~~~~~~~ \end{equation} for all $t \in [ {0,T} ]$, and $A_{2}(t)$ satisfies ${\| {{A_2}( {t} )} \|_{{H^1}( {\mathbb{R}} )}} \le {e^{ - \varepsilon \gamma | {\rho ( t )} |}} + \varepsilon$. \end{lemma} \begin{proof} We have \begin{align*} I &= ( {1 - \lambda \partial _x^2} ){R_t} + {( {{R_{xx}} - R + {m_\varepsilon }{R^2}} )_x} + \varepsilon n_0'( {\varepsilon x} ){Q_{c( t )}}R \\ & = ( {1 - \lambda \partial _x^2} )\frac{{( { \wedge {Q_c}{c'}\varepsilon - Q_c'c} ) {m( {\varepsilon \rho } )} - {Q_c}{{m'( {\varepsilon \rho } )}}\varepsilon c}} {{{{m^2( {\varepsilon \rho } )}}}} + \frac{1}{{m( {\varepsilon \rho } )}}Q_c'''\\ &\quad - \frac{1}{m( {\varepsilon \rho } )}Q_c' + \frac{1}{{{m^2( {\varepsilon \rho } )}}}{( {m( {\varepsilon x} )Q_c^2} )_x} + \frac{1}{m( {\varepsilon \rho } )}\varepsilon n_0'( {\varepsilon x} )Q_c^2. \end{align*} By the Taylor expansion, we obtain \begin{align*} {( {m( {\varepsilon x} )Q_c^2} )_x} &= {( {m( {\varepsilon \rho ( t )} )Q_c^2} )_x} + \varepsilon {m'}( {\varepsilon \rho ( t )} ){( {yQ_c^2} )_x}\\ &\quad + \frac{1}{2}{\varepsilon ^2}{m''}( {\varepsilon \rho ( t )} ) {( {{y^2}Q_c^2} )_x}+ {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^3}} ) \end{align*} and $$ n_0'( {\varepsilon x} )Q_c^2 = n_0'( {\varepsilon \rho ( t )} )Q_c^2 + \varepsilon n_0''( {\varepsilon \rho ( t )} )yQ_c^2 + {O_{{H^1}({\mathbb{R}})}}( {{\varepsilon ^2}} ), $$ which implies \begin{align*} I &=( {1 - \lambda \partial _x^2} )\frac{{( { \wedge {Q_c}{c'}\varepsilon - Q_c'c} ){m( {\varepsilon \rho } )} - {Q_c}{m'{( {\varepsilon \rho } )}}\varepsilon c}}{{{m^2 {( {\varepsilon \rho } )}}}} \\ &\quad + \frac{1}{{{m^2}{( {\varepsilon \rho } )}}} \big[ {{{( {m{( {\varepsilon \rho } )}Q_c^2} )}_x} + \varepsilon {m'{( {\varepsilon \rho } )}}{{( {yQ_c^2} )}_x} + \frac{1}{2}{\varepsilon ^2}{m''{( {\varepsilon \rho } )}} {{( {{y^2}Q_c^2} )}_x}} \big] \\ &\quad + \frac{Q_c'''}{m{( {\varepsilon \rho } )}} - \frac{Q_c'}{m{( {\varepsilon \rho } )}} + \frac{1}{m{( {\varepsilon \rho } )}}\varepsilon \big( {n_0'{( {\varepsilon \rho } )}Q_c^2 + \varepsilon n_0''{( {\varepsilon \rho } )}yQ_c^2} \big) +{O_{{H^1}({\mathbb{R}} )}}( {{\varepsilon ^3}} ). \end{align*} Then, using $( {1 + \lambda c} )Q_c'' + Q_c^2 = ( {1 + c} ){Q_c}$, we obtain \begin{align*} I &=\frac{1}{m{( {\varepsilon \rho } )}}{[ {Q_c^2 - ( {1 + c} ){Q_c} + ( {1 + \lambda c} )Q_c''} ]_x} \\ &\quad +\varepsilon \Big[ \frac{{{c'}}}{m{( {\varepsilon \rho } )}} ( {1 - \lambda \partial _x^2} ) \wedge {Q_c} - \frac{{{m'{( {\varepsilon \rho } )}}c}}{{{m^2{( {\varepsilon \rho } )}}}} ( {1 - \lambda \partial _x^2} ){Q_c} + \frac{{{m'{( {\varepsilon \rho } )}}}}{{{m^2}{( {\varepsilon \rho } )}}} {{( {yQ_c^2} )}_x} \\ &\quad + \frac{{n_0'{( {\varepsilon \rho } )}}} {m{( {\varepsilon \rho } )}}Q_c^2 \Big] +{\varepsilon ^2}\Big[ {\frac{{{m''{( {\varepsilon \rho } )}}}} {{2{m^2{( {\varepsilon \rho } )}}}}{{( {{y^2}Q_c^2} )}_x} + \frac{{n_0''{( {\varepsilon \rho } )}}}{m{( {\varepsilon \rho } )}}yQ_c^2} \Big] + {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^3}} ) \\ &=\varepsilon \Big[ \frac{{{c'}}}{m{( {\varepsilon \rho } )}} ( {1 - \lambda \partial _x^2} ) \wedge {Q_c} - \frac{{{m'{( {\varepsilon \rho } )}}c}}{{{m^2{( {\varepsilon \rho } )}}}} ( {1 - \lambda \partial _x^2} ){Q_c} + \frac{{{m'{( {\varepsilon \rho } )}}}} {{{m^2}{( {\varepsilon \rho } )}}}{{( {yQ_c^2} )}_x} \\ &\quad + \frac{{n_0'{( {\varepsilon \rho } )}}}{m{( {\varepsilon \rho } )}}Q_c^2 \Big] +{\varepsilon ^2}\Big[ {\frac{{{m''{( {\varepsilon \rho } )}}}} {{2{m^2{( {\varepsilon \rho } )}}}}{{( {{y^2}Q_c^2} )}_x} + \frac{{n_0''{( {\varepsilon \rho } )}}}{m{( {\varepsilon \rho } )}}yQ_c^2} \Big] + {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^3}} ). \end{align*} From equation \eqref{e3.12}, we have $$ A_{2}(t,y)\in S({\mathbb{R}}),\quad {\| {{A_2}( {t,y} )} \|_{{H^1}( {\mathbb{R}} )}} \le {e^{ - \varepsilon \gamma | {\rho ( t )} |}} + \varepsilon. $$ This completes the proof. \end{proof} We now consider a linear elliptic operator $L$, for any fixed $c>0$. Let \begin{equation} \label{e3.13} LB=-(1+\lambda c){B}''+(1+c)B-2Q_{c}B, \end{equation} where \begin{equation} \label{e3.14} {{Q}_{c}}(x)={(1+c)}Q\Big( \sqrt{\frac{1+c}{1+\lambda c}}x \Big). \end{equation} \begin{lemma} \label{lem3.3} Assume that ${{V}_{c}}$ satisfies the hypothesis {\rm (H1)-(H2)}. Then \begin{align*} II &=( {1 - \lambda \partial _x^2} )( {{c'}{\partial _c}W - c{\partial _y}W} )-{( {LW} )_y}\\ &\quad +{\varepsilon ^2}\big[ {( {1 - \lambda \partial _x^2} ) \big( {n_0''( {\varepsilon \rho } ){c}{V_c} + n_0'( {\varepsilon \rho } ){c'}{\partial _c}{V_c}} \big)} \big]+{\varepsilon ^2}{A_3}( {t,y} ), \end{align*} where \begin{align*} {A_3}( {t,y} ) &=( {1 - \lambda \partial _x^2} )n_0'( {\varepsilon \rho } ){\partial _t}{V_c} + \frac{{2{m'{( {\varepsilon \rho } )}}}}{m{( {\varepsilon \rho } )}} {{( {y{Q_c}n_0'{( {\varepsilon \rho } )}{V_c}} )}_x} \\ &\quad + {O_{{H^1}( {\mathbb{R}} )}}( {{e^{ - \varepsilon \gamma | {\rho ( t )} |}}} ). \end{align*} \end{lemma} \begin{proof} Let ${{B}_{c}}(t,y)={W}$ be a smooth function with $y=x-\rho (t)$, then by using $$ II(B)= ( {1 - \lambda \partial _x^2} ){B_t} + {( {{B_{xx}} - B + 2{m_\varepsilon }RB} )_x} + \varepsilon n_0'( {\varepsilon x} ){Q_{c( t )}}B $$ we obtain \begin{align*} II(B)&= ( {1 - \lambda \partial _x^2} )[ {{c'}{\partial _c}B + {B_t} - ( {{\rho '} - c} ){B_y}} ] \\ &\quad + {\Big[ {{B_{yy}} - ( {1 - \lambda \partial _x^2} )cB - B + 2{Q_{c( t )}}B + \frac{{2{m'{( {\varepsilon \rho } )}}}}{m{( {\varepsilon \rho } )}} \varepsilon y{Q_{c( t )}}B} ]_x}\\ &\quad + O( {\varepsilon n_0'( {\varepsilon x} ){Q_{c( t )}}B} )\\ &=( {1 - \lambda \partial _x^2} )( {{c'}{\partial _c}B + {B_t}} ) + {[ {( {1 + \lambda c} ){B_{xx}} - ( {1 + c} )B + 2{Q_{c( t )}}B} ]_x} \\ &\quad + \frac{{2{m'{( {\varepsilon \rho } )}}}}{m{( {\varepsilon \rho } )}} \varepsilon {( {y{Q_{c( t )}}B} )_x} + {O_{{H^1}( {\mathbb{R}} )}} \big( {{\varepsilon ^2}{e^{ - \varepsilon \gamma | {\rho ( t )} |}}} \big). \end{align*} Applying the identity $W( {t,x} ) = \varepsilon n_0'( {\varepsilon \rho ( t )} ){V_{c( t )}} ( {\varepsilon t,y} )$ defined in \eqref{e3.5}, we have \begin{gather*} {\partial _c}W = \varepsilon n_0'( {\varepsilon \rho ( t )} ){\partial _c}{V_c},\\ {W_t} = \varepsilon \big[ {\varepsilon n_0''( {\varepsilon \rho ( t )} ){\rho '} ( t ){V_c} + n_0'( {\varepsilon \rho ( t )} ) \big( {{c'}\varepsilon {\partial _c}{V_c} + \varepsilon {\partial _t}{V_c} - c{\partial _y}{V_c}} \big)} \big], \end{gather*} and \begin{align*} II(W)&=( {1 - \lambda \partial _x^2} )( {{c'}{\partial _c}W + {W_t}} )-{( {LW} )'} + \frac{{2{m'( {\varepsilon \rho } )}}}{m( {\varepsilon \rho } )} \varepsilon {( {y{Q_{c( t )}}W} )_x} \\ &\quad + {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^2} {e^{ - \varepsilon \gamma | {\rho ( t )} |}}} )\\ &={\varepsilon ^2}\Big[ ( {1 - \lambda \partial _x^2} ) ( {n_0''( {\varepsilon \rho } ){\rho '}{V_c} + n_0'( {\varepsilon \rho } ){c'} {\partial _c}{V_c} + n_0'( {\varepsilon \rho } ){\partial _t}{V_c}} ) \\ &\quad + \frac{{2{m'( {\varepsilon \rho } )}}}{m( {\varepsilon \rho } )} {{( {y{Q_c}n_0'( {\varepsilon \rho } ){V_c}} )}_x} \Big] + \varepsilon n_0'( {\varepsilon \rho } )( {1 - \lambda \partial _x^2} ) ( {{c'}{\partial _c}{V_c} - c{\partial _y}{V_c}} ) \\ &\quad - \varepsilon n_0'( {\varepsilon \rho } ){( {L{V_c}} )_y} + {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^2} {e^{ - \varepsilon \gamma | {\rho ( t )} |}}} ) \\ &=( {1 - \lambda \partial _x^2} )( {{c'}{\partial _c}W - c{\partial _y}W} ) -{( {LW} )_y}+{\varepsilon ^2} \big[ ( {1 - \lambda \partial _x^2} )\big( n_0''( {\varepsilon \rho } ){c}{V_c} \\ &\quad + n_0'( {\varepsilon \rho } ){c'}{\partial _c}{V_c} \big) \big] +{\varepsilon ^2}\Big[ ( {1 - \lambda \partial _x^2} ) n_0'( {\varepsilon \rho } ){\partial _t}{V_c} + \frac{{2{m'( {\varepsilon \rho } )}}}{m( {\varepsilon \rho } )} {{( {y{Q_c}n_0'( {\varepsilon \rho } ){V_c}} )}_x}\\ &\quad + {O_{{H^1}( {\mathbb{R}} )}}( {{e^{ - \varepsilon \gamma | {\rho ( t )} |}}} ) \Big]. \end{align*} This completes the proof. \end{proof} \begin{lemma} \label{lem3.4} For any $t \in [ {0,T} ]$, $$ III = {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^2} {e^{ - \varepsilon \gamma | {\rho ( t )} |}}} ). $$ \end{lemma} \begin{proof} Recall that $III := {( {{m_\varepsilon }{W^2}} )_x} $. Then we obtain $$ III= {\varepsilon ^2}{\big( {n_0'( {\varepsilon \rho } )} \big)^2} {( {m( {\varepsilon x} )V_c^2} )_x} = {\varepsilon ^2}{( {n_0'( {\varepsilon \rho } )} )^2} \big[ {\varepsilon {m'}( {\varepsilon x} )V_c^2 + m( {\varepsilon x} ){{( {V_c^2} )}'}} \big]. $$ Since ${V_c}$ satisfies the hypothesis (H1)-(H2), ${( {V_c^2} )'}\in S({\mathbb{R}})$ holds. By taking the space derivative, we can see the desired result. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3.1}] According to the estimates from Lemmas \ref{lem3.2}, \ref{lem3.3} and \ref{lem3.4}, we obtain \begin{equation} \label{e3.15} S[ K ] = ( {1 - \lambda \partial _x^2} )( {{c'}{\partial _c}K - c{\partial _y}K} ) + {S_0}[ K ], \end{equation} where \begin{equation} \label{e3.16} \begin{aligned} {S_0}[ K ] &= \varepsilon [ {{A_1}( {t,y} ) - n_0'( {\varepsilon \rho }) {{( {L{V_c}} )}_y}} ] \\ &\quad + {\varepsilon ^2}\big[ {( {1 - \lambda \partial _x^2} ) ( {n_0''( {\varepsilon \rho })c{V_c} + n_0'( {\varepsilon \rho } ){c'}{\partial _c}{V_c} + n_0'( {\varepsilon \rho } ){\partial _t}{V_c}} )} \big] \\ &\quad + {\varepsilon ^2}\big[ {\frac{{{m''}( {\varepsilon \rho } )}}{{2{m^2} ( {\varepsilon \rho } )}}{{( {{y^2}Q_c^2} )}_x} + \frac{{n_0''( {\varepsilon \rho } )}}{{m( {\varepsilon \rho } )}}yQ_c^2 + \frac{{2{m'( {\varepsilon \rho } )}}} {{m( {\varepsilon \rho } )}}{{( {y{Q_c}n_0'( {\varepsilon \rho } ){V_c}} )}_x}} \big] \\ &\quad + {\varepsilon ^2}{O_{{H^1}( {\mathbb{R}} )}} ( {{e^{ - \varepsilon \gamma | {\rho ( t )} |}} + \varepsilon } ). \end{aligned} \end{equation} The next step is the resolution of the linear differential equation about the first order term in $\varepsilon$. From \eqref{e3.16}, we want to solve \begin{equation} \label{e3.20} n_0'( {\varepsilon \rho ( t )} ){( {L{V_c}} )_y} = {A_1}( {t,y} ). \end{equation} Obviously, accuracy of the error term can be raised to $O(\varepsilon^{2})$ now. It is not difficult to check that, for all $y \in {\mathbb{R}}$ and $t$ fixed, ${A_1}$ satisfies the regularity conditions: \begin{equation} \label{e3.21} \int_{\mathbb{R}} {{A_1}} ( {t,y} ){Q_c}( y )dy = 0. \end{equation} \begin{lemma} \label{lem3.5} The operator $L$ defined on ${L^2}( {\mathbb{R}} )$ by \eqref{e3.13} satisfies \begin{itemize} \item[(1)] The kernel of $L$ is $Q_c'$, that is $LQ_c' = 0$. \item[(2)] For any $f = f( y ) \in {L^2}( {\mathbb{R}} )$ which satisfies $\int_{\mathbb{R}} {fQ_c'}(y)dy = 0$, and there exists a unique function ${f_0}$ such that $\int_{\mathbb{R}} {{f_0}} Q_c'(y)dy = 0$ and $L{f_0} = f$. Furthermore, if $f$ is odd, then $f_0$ is odd. \end{itemize} \end{lemma} Set $c > 0$ and define \begin{equation} \label{e3.22} \psi ( x ): = - \frac{{{Q'}( x )}}{{Q( x )}},\quad {\psi _c}( x ): = - \frac{{Q_c'( x )}}{{{Q_c}( x )}} = \sqrt {\frac{{1 + c}}{{1 + \lambda c}}} \psi ( {\sqrt {\frac{{1 + c}}{{1 + \lambda c}}} x} ). \end{equation} A direct computation yields \begin{equation} \label{e3.23} \lim _{x \to - \infty } \psi ( x ) = - 1,\quad \lim _{x \to + \infty } \psi ( x ) = 1. \end{equation} \begin{lemma} \label{lem3.6} There exists a unique solution ${V_c} = {V_{c( t )}}( {t,y} )$ satisfying $$ n_0'( {\varepsilon \rho} ){( {L{V_c}} )_y} = {A_1}( {t,y} ) $$ such that, for every $t$, \begin{equation} \label{e3.24} {V_c}( {t,y} ): = {\alpha _c}( t )\Big( {{\psi _c}( y ) - \sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big) + {\beta _c}( t )Q_c'( y ) + {V_1}( {t,y} ) + {\sigma _c}( t ), \end{equation} and \begin{equation} \label{e3.25} \lim _{y \to - \infty } {V_c}( {t,y} ) =- 2\sqrt {\frac{{1 + c}}{{1 + \lambda c}}}{\alpha_c}( t );\quad | {{V_c}( y )} | \le k{e^{ - \gamma y}},\,\,\,as \,\,y \to +\infty, \end{equation} with ${V_1}( y ) \in \emph{S(R)}$ for all $t,{\alpha _c}( t ),{\beta _c}( t )$ and ${\sigma _c}( t ) \in {\mathbb{R}}$. Moreover, we have \begin{equation} \label{e3.26} {\alpha _c}( t ): = \frac{1}{{2n_0'( {\varepsilon \rho ( t )} )}} \sqrt {\frac{{1 + \lambda c}}{{1 + c}}} \int_{\mathbb{R}} {{A_1}} ( {t,y} )dy \ne 0. \end{equation} \end{lemma} \begin{proof} The proof of this lemma is divided into three steps. \smallskip \noindent\textbf{Step 1.} The first step is to prove the existence of $n_0'( {\varepsilon \rho ( t )} ){( {L{V_c}} )_y} = {A_1}( {t,y} )$, where ${V_c}$ was established in \cite{s20}. For equation \eqref{e3.24}, we have $$ L{V_c( y )} = {\alpha _c}( t ) L\Big( {{\psi _c}( y ) - \sqrt {\frac{{1 + c}} {{1 + \lambda c}}} } \Big) + {\beta _c}( t ) LQ_c'( y ) + L{V_1( y )} + L{\sigma _c}( t ). $$ So $$ L{V_1( y )} = H( y ) - {\alpha _c}( t )L\Big( {{\psi _c}( y ) - \sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big) - {\theta _c}( t ), $$ with $$ H( y ) = \frac{1}{{n_0'( {\varepsilon \rho } )}}\int_{\mathbb{R}} {{A_1}} ( {t,y} )dy, \quad {\theta _c}( t ) = L{\sigma _c}( t ). $$ Without lose of generality, we assume $$ {\theta _c}( t ) = 2{\alpha _c}( t ) {\sqrt {\frac{{1 + c}}{{1 + \lambda c}}}}. $$ The solvability of equation \eqref{e3.20} is equivalent to \begin{align*} &\int_{\mathbb{R}} {L{V_1( y )}} Q_c'( {y} )dy \\ &= \int_{\mathbb{R}} {\Big[ {H( y ) - {\alpha _c}L\Big( {{\psi _c}( y ) - \sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big) - 2{\alpha _c} {\sqrt {\frac{{1 + c}}{{1 + \lambda c}}} }} \Big]} Q_c'( {y} )dy \\ & = \int_{\mathbb{R}} {\Big[ {H( y ) - {\alpha _c}\Big( {L\Big( {{\psi _c}( y ) - \sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big) + 2\sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big)} \Big]} Q_c'( {y} )dy \\ &= \int_{\mathbb{R}} {H( y )} Q_c'( y )dy = - \int_{\mathbb{R}} {{Q_c}} ( y )dH \\ &= - \frac{1}{{n_0'( {\varepsilon \rho } )}}\int_{\mathbb{R}} {{Q_c}} ( y ){A_1}( y )dy = 0. \end{align*} Since $LQ_c'( {y} ) = 0$ and $\int_{\mathbb{R}} {L{V_1}}( {y} ) Q_c'( {y} )dy = 0$, according to Lemma 3.5, there exists a function ${V_1}( y )$ satisfying $\int_{\mathbb{R}} {{V_1}( {y} )} Q_c'( {y} )dy = 0$. Note that $$ \lim _{y \to - \infty } \Big[ {H( y ) - {\alpha _c}\Big( {L\Big( {{\psi _c}( y ) - \sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big) + 2\sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big)} \Big] = 0 $$ and \begin{align*} &\lim _{y \to + \infty } \Big[ {H( y ) - {\alpha _c}\Big( {L\Big( {{\psi _c}( y ) - \sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big) + 2\sqrt {\frac{{1 + c}}{{1 + \lambda c}}} } \Big)} \Big]\\ &= \frac{1}{{n_0'( {\varepsilon \rho } )}}\int_{\mathbb{R}} {{A_1}}( y )dy - 2{\alpha _c}\sqrt {\frac{{1 + c}}{{1 + \lambda c}}} =0, \end{align*} So we have ${V_1}( y ) \in \emph{S(R)}$ and $$ {\alpha _c} = \frac{1}{{2n_0'( {\varepsilon \rho } )}} \sqrt {\frac{{1 + \lambda c}}{{1 + c}}} \int_{\mathbb{R}} {{A_1}} ( {t,y} )dy. $$ \smallskip \noindent\textbf{Step 2.} In this step we show that \begin{align*} {\alpha _c} &= \frac{{\int_{\mathbb{R}} {{Q_c}} ( y )dy }} {{2n_0'( {\varepsilon \rho } )}}\sqrt {\frac{{1 + \lambda c}}{{1 + c}}} \Big[ \frac{{1 + 2\lambda c + \lambda }}{{2( {1 + \lambda c} )( {1 + c} )}} \frac{{{c'}}}{m( {\varepsilon \rho } )} - \frac{{{m'( {\varepsilon \rho } )}c}} {m( {\varepsilon \rho } )} \\ &\quad + \frac{{n_0'( {\varepsilon \rho } )( {1 + c} )}} {m( {\varepsilon \rho } )} \Big]. \end{align*} From equation \eqref{e3.11}, one has \begin{align*} {A_1}( {t,y} ) &= \frac{{{c'}}}{{m( {\varepsilon \rho } )}}( {1 - \lambda \partial _x^2} ) \wedge {{{Q_{c}}( y )}} - \frac{{{m'}( {\varepsilon \rho } )c}}{{{m^2} ( {\varepsilon \rho } )}}( {1 - \lambda \partial _x^2} ){{{Q_{c}}( y )}} \\ &\quad + \frac{{{m'}( {\varepsilon \rho } )}}{{{m^2}( {\varepsilon \rho } )}} {( {y{Q^2_{c}}( y )} )_x} + \frac{{n_0'( {\varepsilon \rho } )}}{{m( {\varepsilon \rho } )}}{Q^2_{c}}( y ). \end{align*} Next we consider the following four integrals: \[ \int_{\mathbb{R}} {( {1 - \lambda \partial _x^2} )} \wedge {Q_c}( y )dy,\quad \int_{\mathbb{R}}{( {1 - \lambda \partial _x^2} )} {Q_c}( y )dy,\quad \int_{\mathbb{R}}{( {yQ_c^2( y )} )} _x dy, \quad \int_{\mathbb{R}} {Q_c^2} ( y )dy. \] Note that \begin{align*} &\int_{\mathbb{R}} {( {1 - \lambda \partial _x^2} )} \wedge {Q_c}( y )dy\\ &= \frac{1}{{1 + c}}\int_{\mathbb{R}} {( {1 - \lambda \partial _x^2} )} \Big( {{Q_c}( y ) + \frac{{1 - \lambda }}{{2( {1 + \lambda c} )}}yQ_c'( y )} \Big)dy \\ &= \frac{1}{{1 + c}}[ {\int_{\mathbb{R}} {\big( {{Q_c}( y ) - \lambda Q_c''( y )} \big)} dy + \int_{\mathbb{R}} {\Big( {\frac{{1 - \lambda }}{{2( {1 + \lambda c} )}}yQ_c'( y ) - \frac{{\lambda ( {1 - \lambda } )}}{{2( {1 + \lambda c} )}}{{( {yQ_c'( y )} )}''}} \Big)dy} } ] \\ &= \frac{1}{{1 + c}}\Big[ \int_{\mathbb{R}} {{Q_c}( y )} dy - \frac{{1 - \lambda }}{{2( {1 + \lambda c} )}}\int_{\mathbb{R}} {Q_c}( y )dy \\ &\quad - \frac{{\lambda ( {1 - \lambda } )}}{{2( {1 + \lambda c} )}} \int_{\mathbb{R}} {{{( {yQ_c'}( y ) )}''}dy} \Big]\\ &= \big[ {\frac{1}{{1 + c}} - \frac{{1 - \lambda }}{{2( {1 + \lambda c} ) ( {1 + c} )}}} \big]\int_{\mathbb{R}} {{Q_c}} ( y )dy, \end{align*} \begin{gather*} \int_{\mathbb{R}} {( {1 - \lambda \partial _x^2} )} {Q_c}( y )dy = \int_{\mathbb{R}} {{Q_c}} ( y )dy - \lambda \int_{\mathbb{R}} {Q_c''} ( y )dy = \int_{\mathbb{R}} {{Q_c}} ( y )dy,\\ \int_{\mathbb{R}} {( {yQ_c^2( y )} )} _x\,dy = 0. \end{gather*} According to $$ r\int_{\mathbb{R}} {Q_c^r} ( y )dy = \frac{{2r + 1}}{{3( {1 + c} )}}\int_{\mathbb{R}} {Q_c^{r + 1}} ( y )dy, $$ when $r = 1$ one has $$ \int_{\mathbb{R}} {Q_c^2} ( y )dy = ( {1 + c} )\int_{\mathbb{R}} {{Q_c}} ( y )dy. $$ So we can get \begin{align*} {\alpha _c} &= \frac{{\int_{\mathbb{R}} {{Q_c}} ( y )dy }}{{2n_0' ( {\varepsilon \rho } )}}\sqrt {\frac{{1 + \lambda c}}{{1 + c}}} \Big[ \frac{{1 + 2\lambda c + \lambda }}{{2( {1 + \lambda c} )( {1 + c} )}} {\frac{{{c'}}}{m( {\varepsilon \rho } )} - \frac{{{m'( {\varepsilon \rho } )}c}} {m( {\varepsilon \rho } )} + \frac{{n_0'( {\varepsilon \rho } )( {1 + c} )}} {m( {\varepsilon \rho } )}} \Big]. \end{align*} \smallskip \noindent\textbf{Step 3.} From step 1, we have $$ {\theta _c}( t ) = 2{\alpha _c}( t ) {\sqrt {\frac{{1 + c}}{{1 + \lambda c}}} }. $$ By computing ${\theta _c}( t ) =2{\alpha _c}( t ){\sqrt {\frac{{1 + c}}{{1 + \lambda c}}}}$, we can easily assert that $V_{c}$ is exponential decay as $y\to+\infty$, that is $\lim _{y \to + \infty } {V_c} = 0$. This completes the proof of the lemma. \end{proof} We use a method similarly to the one in \cite{s22}. According to \eqref{e3.25}, we obtain the estimation \eqref{e3.8}. In addition, from Lemma 3.6, we see that $V_{c}$ satisfies the hypothesis (H1)-(H2). From \eqref{e3.16} we arrive at \eqref{e3.9}. All these complete the proof of Theorem \ref{thm3.1}. \end{proof} \section{Solution for $K \in L^{2}(R)$} The following method is similar to those introduced in \cite{s20,s22,s23}. Since $K \notin {L^2}( {\mathbb{R}} )$, we introduce a cut-off function $\zeta \in {C^\infty }( {\mathbb{R}} )$ satisfying the following properties \begin{equation} \label{e4.1} \begin{gathered} \zeta ( y ) \equiv \begin{cases} 0 &\text{for }y \le - 1,\\ 1 &\text{for } y \ge 1,\end{cases} \\ 0 \le \zeta ( y ) \le 1,\quad 0 \le {\zeta '}( y ) \le 1\quad\text{for } y \in {\mathbb{R}}. \end{gathered} \end{equation} Similarly, for $R( {t,x} )$ and $W( {t,x} )$ constructed in equation \eqref{e3.5}, we define a new approximate solution $\widetilde K$: \begin{equation} \label{e4.2} \widetilde K: = {\zeta _\varepsilon }( y )K( {t,x} ) = {\zeta _\varepsilon }( y )( {R( {t,x} ) + W( {t,x} )} ), \end{equation} where \begin{equation} \label{e4.3} {\zeta _\varepsilon }( y ): = \zeta ( {\varepsilon y + 2} ). \end{equation} \begin{theorem} \label{thm4.1} For all $0 < \varepsilon < \tau$, there exist positive constants $\tau$ and $ k$ such that: (1) \begin{equation} \label{e4.4} \begin{gathered} \widetilde K = 0\quad \text{for } y \le - \frac{3}{\varepsilon },\\ \widetilde K = {K( {t,x} )}\quad \text{for } y \ge - \frac{1}{\varepsilon }. \end{gathered} \end{equation} For any $t$ in a given interval, $W( {t,x} ) \in {H^1}( {\mathbb{R}} )$ satisfies \begin{equation} \label{e4.5} {\| {W( {t,x} )} \|_{{H^1}( {\mathbb{R}} )}} \le k{\varepsilon ^{1/2}} {e^{ - \gamma \varepsilon | {\rho ( t )} |}}. \end{equation} (2) The error term associated to the new solution $\widetilde K$ satisfies \begin{equation} \label{e4.6} \begin{aligned} S[ {\widetilde K} ] &= ( {1 - \lambda \partial _x^2} )[ {c'}( {{\partial _c}\widetilde K + {O_{{H^1} ( {\mathbb{R}})}}( {\varepsilon n_0'} )} ) \\ &\quad - c\big( {{\partial _y}\widetilde K + {O_{{H^1}({\mathbb{R}} )}} ( {\varepsilon n_0'} )} \big) ] + {S_0}[ {\widetilde K} ], \end{aligned} \end{equation} where \begin{equation} \label{e4.7} {\| {{S_0}[ {\widetilde K} ]} \|_{{H^1}( {\mathbb{R}} )}} \le k{\varepsilon ^{3/2}}{e^{ - \gamma \varepsilon | {\rho ( t )} |}}. \end{equation} \end{theorem} \begin{proof} (1) It follows from \eqref{e4.5} that $$ {\| {{\zeta _\varepsilon }( y )W( {t,x} )} \|_{{H^1}( {\mathbb{R}} )}} \le k{\| {W( {t,x} )} \|_{{H^1}( {y \ge - \frac{3}{\varepsilon }} )}}. $$ From (H1)-(H2), we have $$ {\| {\varepsilon n_0'( {\varepsilon \rho ( t )} ){V_{c( t )}}( y )} \|_{{H^1}( {y \ge - \frac{3}{\varepsilon }} )}} \le k{\varepsilon ^{1/2}} {e^{ - \gamma \varepsilon | {\rho ( t )} |}}. $$ (2) We make a simple calculation as follows \begin{align*} S[ {\widetilde K} ] = S[ {{\zeta _\varepsilon }( y )K} ] &= S[ {{\zeta _\varepsilon }( y )( {R + W} )} ]\\ &= {\zeta _\varepsilon }S[ K ] + {( {{\zeta _\varepsilon }} )_t} ( {1 - \lambda \partial _x^2} )K + 2\varepsilon {\zeta _\varepsilon }'{K_{xx}} + 3{\varepsilon ^2}{\zeta _\varepsilon }''{K_x} + {\varepsilon ^3}{\zeta _\varepsilon }'''K. \end{align*} and $$ 2\varepsilon {\zeta _\varepsilon }'{K_{xx}} + 3{\varepsilon ^2} {\zeta _\varepsilon }''{K_x} + {\varepsilon ^3}{\zeta _\varepsilon }'''K = {O_{{H^1}( {\mathbb{R}})}}( {{\varepsilon ^{3/2}}{e^{ - \gamma \varepsilon | {\rho ( t )} |}}} ) + {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^{30}}} ). $$ Similarity, we have $$ {( {{\zeta _\varepsilon }} )_t}( {1 - \lambda \partial _x^2} )K = {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^{3/2}} {e^{ - \gamma \varepsilon | {\rho ( t )} |}}} ) + {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^{30}}} ), $$ so we obtain $$ S[ {{\zeta _\varepsilon }( y )K} ] = {\zeta _\varepsilon }S[ K ] + {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^{3/2}}{e^{ - \gamma \varepsilon | {\rho ( t )} |}}} ) + {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^{30}}} ). $$ Finally, from (H1)-(H2), and \eqref{e3.5} and \eqref{e3.7}, we have $$ {\| {{\zeta _\varepsilon }{S_0}[ { K} ]} \|_{{H^1}( {\mathbb{R}})}} \le k{\varepsilon ^{3/2}}{e^{ - \gamma \varepsilon | {\rho ( t )} |}} + {\varepsilon ^3}, $$ \begin{align*} &{\zeta _\varepsilon }( {1 - \lambda \partial _x^2} ) ( {{c'}{\partial _c}K - c{\partial _y}K} ) \\ &= ( {1 - \lambda \partial _x^2} ){c'}{\partial _c}( {{\zeta _\varepsilon }K} ) - ( {1 - \lambda \partial _x^2} )c{\partial _y}( {{\zeta _\varepsilon }K} ) - \varepsilon ( {1 - \lambda \partial _x^2} )c{\zeta _\varepsilon }'K. \end{align*} Note that $$ \varepsilon ( {1 - \lambda \partial _x^2} ){\zeta _\varepsilon }'K = {O_{{H^1}( {\mathbb{R}} )}}( {{\varepsilon ^{3/2}} {e^{ - \gamma \varepsilon | {\rho ( t )} |}}} ). $$ Consequently, we obtain the desired results. \end{proof} \subsection*{Acknowledgements} This work was supported by the National Nature Science Foundation of China (Grant Nos. 11501253, 11371175, 11571141 and 11571140), by the Nature Science Foundation of Jiangsu Province (Grant No. BK 20140525), and by the Advanced Talent of Jiangsu University (Grant No. 14JDG070, 15JDG079). \begin{thebibliography}{00} \bibitem{s4} J. 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