Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

Mohamed Ali Hamza¹, Yuta Wakasugi^2∗ and Shuji Yoshikawa³ [email protected] [email protected] [email protected] ¹ Basic Sciences Department, Deanship of Preparatory Year and Supporting Studies, P. O. Box 1982, Imam Abdulrahman Bin Faisal University, Dammam, KSA. ² Laboratory of Mathematics, Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan ³ Division of Mathematical Sciences, Faculty of Science and Technology, Oita University, Oita, 870-1192, Japan

Abstract.

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients:

\partial_{t}^{2}u+b(t)\partial_{t}u-a(t)\partial_{x}^{2}u+\partial_{x}^{4}u=\partial_{x}\left(N(\partial_{x}u)\right).

In the authors’ previous article [17], the asymptotic profile of solutions for linearized problem ( $N\equiv 0$ ) was classified depending on the assumptions for the coefficients $a(t)$ and $b(t)$ and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: $b(t)\partial_{t}u-a(t)\partial_{x}^{2}u=0$ asymptotically as $t\to\infty$ .

Key words and phrases:

Nonlinear damped beam equations; asymptotic behavior; global existence; variable coefficients

⁰⁰02010 Mathematics Subject Classification. 35G25; 35B40; 35A01⁰⁰0

*

Corresponding author

1. Introduction

We study the Cauchy problem of nonlinear damped beam equation

\displaystyle\left\{\begin{aligned} &\partial_{t}^{2}u+b(t)\partial_{t}u-a(t)\partial_{x}^{2}u+\partial_{x}^{4}u=\partial_{x}\left(N(\partial_{x}u)\right),&\qquad&t\in(0,\infty),x\in\mathbb{R},\\ &u(0,x)=u_{0}(x),\ \partial_{t}u(0,x)=u_{1}(x),&\qquad&x\in\mathbb{R},\end{aligned}\right.

(1.1)

where $u=u(t,x)$ is a real-valued unknown, $a(t)$ and $b(t)$ are given positive functions of $t$ , $N(\partial_{x}u)$ denotes the nonlinear function, and $u_{0}$ and $u_{1}$ are given initial data.

Before giving more precise assumptions for $a(t)$ , $b(t)$ and $N$ and our result, we first mention the physical background and the mathematical motivations of the problem. The equation (1.1) corresponds to the so-called Falk model under isothermal assumption with the damping term. The Falk model is one of the models for a thermoelastic deformation with austenite-martensite phase transitions on shape memory alloys:

\begin{cases}\partial_{t}^{2}u+\partial_{x}^{4}u=\partial_{x}\left\{(\theta-\theta_{c})\partial_{x}u-(\partial_{x}u)^{3}+(\partial_{x}u)^{5}\right\},\qquad&\\ \partial_{t}\theta-\partial_{x}^{2}\theta=\theta\partial_{x}u\partial_{t}\partial_{x}u,\qquad&t\in(0,\infty),x\in\mathbb{R},\\ u(0,x)=u_{0}(x),\ \partial_{t}u(0,x)=u_{1}(x),\ \theta(0,x)=\theta_{0}(x),\qquad&x\in\mathbb{R},\end{cases}

where $u$ and $\theta$ are the displacement and the absolute temperature, respectively, and $\theta_{c}$ is a positive constant representing the critical temperature for the phase transition. If we assume the temperature are controllable and uniformly distributed with respect to the space, that is, $\theta$ is given function uniform in $x$ such as $\theta-\theta_{c}=a(t)$ and set $N(\varepsilon)=\varepsilon^{5}-\varepsilon^{3}$ , then the problem (1.1) is surely derived. Our interest directs to the behavior of solution around the initial temperature $\theta_{0}$ is closed to the critical temperature $\theta_{c}$ . Indeed, the Lyapunov stability for the solution of the Falk model is shown in [11], which claims that the temperature tends to the function uniformly distributed in $x$ in the bounded domain case. For more precise information of the Falk model of shape memory alloys, we refer the reader to Chapter 5 in [2]. We are also motivated by the extensible beam equation proposed by Woinovsky-Krieger [16]:

\partial_{t}^{2}u-\left(\int_{\mathbb{R}}|\partial_{x}u|^{2}dx\right)\partial_{x}^{2}u+\partial_{x}^{4}u=0.

(1.2)

In [1], the model with the damping term was proposed and the stability result was shown. The problem (1.1) corresponds to the nonlinear generalization for the equation. As the observation similar to the Kirchhoff equation, the linearized problem substituting the given function $a(t)$ into the nonlocal term was also studied by e.g. [4], [10], [17] and [6].

Next, let us explain the mathematical background of our problem. It is well-known that the solution of the Cauchy problem for the damped wave equation $\partial_{t}^{2}u+\partial_{t}u-\partial_{x}^{2}u=0$ behaves as the solution for the heat equation $\partial_{t}u-\partial_{x}^{2}u=0$ asymptotically as $t\to\infty$ (see e.g. [7]). Roughly speaking, this implies that $\partial_{t}^{2}u$ decays faster than $\partial_{t}u$ as $t\to\infty$ . From the same observation, the solution for the beam equation $\partial_{t}^{2}u+\partial_{t}u-\partial_{x}^{2}u+\partial_{x}^{4}u=0$ behaves as the solution for the heat equation $\partial_{t}u-\partial_{x}^{2}u=0$ asymptotically as $t\to\infty$ , because $\partial_{x}^{4}u$ decays faster than $\partial_{x}^{2}u$ . The above observation induces the investigation of the solution for the equation with time variable coefficient: $\partial_{t}^{2}u+b(t)\partial_{t}u-\partial_{x}^{2}u=0$ with $b(t)\sim(1+t)^{\beta}$ . The precise analysis implies that the solution behaves as the solution for the heat equation $b(t)\partial_{t}u-\partial_{x}^{2}u=0$ when $\beta<-1$ , and on the other hand, that the solution behaves as the solution for the wave equation $\partial_{t}^{2}u-\partial_{x}^{2}u=0$ when $-1<\beta<1$ (see e.g. [8], [9], [14] and [15]). Correspondingly, the authors in [17] studied the asymptotic behavior of the solution for the linearized problem of (1.1)

\partial_{t}^{2}u+b(t)\partial_{t}u-a(t)\partial_{x}^{2}u+\partial_{x}^{4}u=0.

As in Figure 1, we divide the two-dimensional regions $\Omega_{j}$ for $(\alpha,\beta)$ ( $j=1,2,3,4,5$ ) by

	$\displaystyle\Omega_{1}$	$\displaystyle:=\left\{(\alpha,\beta)\in\mathbb{R}^{2}\mid-1<\beta<\min\{\alpha+1,2\alpha+1\}\right\},$
	$\displaystyle\Omega_{2}$	$\displaystyle:=\left\{(\alpha,\beta)\in\mathbb{R}^{2}\mid\max\{-1,2\alpha+1\}<\beta<1\right\},$
	$\displaystyle\Omega_{3}$	$\displaystyle:=\left\{(\alpha,\beta)\in\mathbb{R}^{2}\mid\beta<-1<\alpha\right\},$
	$\displaystyle\Omega_{4}$	$\displaystyle:=\left\{(\alpha,\beta)\in\mathbb{R}^{2}\mid\beta<-1,\alpha<-1\right\},$
	$\displaystyle\Omega_{5}$	$\displaystyle:=\left\{(\alpha,\beta)\in\mathbb{R}^{2}\mid\max\{1,\alpha+1\}<\beta\right\}.$

By a scaling argument, the result gives the conjecture for the classification of the asymptotic behavior of the solution:

(1)

In $(\alpha,\beta)\in\Omega_{1}$ , $u(t)$ behaves as the solution for $b(t)u_{t}-a(t)u_{xx}=0$ .
(2)

In $(\alpha,\beta)\in\Omega_{2}$ , $u(t)$ behaves as the solution for $b(t)u_{t}+u_{xxxx}=0$ .
(3)

In $(\alpha,\beta)\in\Omega_{3}$ , $u(t)$ behaves as the solution for $u_{tt}-a(t)u_{xx}=0$ .
(4)

In $(\alpha,\beta)\in\Omega_{4}$ , $u(t)$ behaves as the solution for $u_{tt}+u_{xxxx}=0$ .
(5)

In $(\alpha,\beta)\in\Omega_{5}$ , $u(t)$ behaves as the solution for over damping case.

As a partial answer, the authors proved the effective damping cases (1) and (2) in [17]. Here we shall give the result for the nonlinear problem (1.1) in the case (1).

From now on, we shall give our main result. To state it precisely, we put the following assumptions.

Assumption (A) The coefficients $a(t)$ and $b(t)$ are smooth positive functions satisfying

\displaystyle C^{-1}(1+t)^{\alpha}\leq a(t)\leq C(1+t)^{\alpha},\quad C^{-1}(1+t)^{\beta}\leq b(t)\leq C(1+t)^{\beta}

(1.3)

and

\displaystyle|a^{\prime}(t)|\leq C(1+t)^{\alpha-1},\quad|b^{\prime}(t)|\leq C(1+t)^{\beta-1}

(1.4)

with some constant $C\geq 1$ and the parameters $\alpha,\beta\in\mathbb{R}$ . Moreover, we assume

\displaystyle(\alpha,\beta)\in\Omega_{1}:=\left\{(\alpha,\beta)\in\mathbb{R}^{2}\mid-1<\beta<\min\{\alpha+1,2\alpha+1\}\right\}.

(1.5)

Assumption (N) The function $N(\partial_{x}u)$ is a linear combination of $(\partial_{x}u)^{2}$ and $p$ -th order terms with $p\geq 3$ . More precisely, the function $N$ has the form

\displaystyle N(z)=\mu z^{2}+\tilde{N}(z),

(1.6)

with some $\mu\in\mathbb{R}$ , where $\tilde{N}\in C^{2}(\mathbb{R})$ satisfies

\displaystyle\tilde{N}^{(j)}(0)=0\quad\text{and}\quad|\tilde{N}^{(j)}(z)-\tilde{N}^{(j)}(w)|\leq C(|z|+|w|)^{p-1-j}|z-w|\quad(z,w\in\mathbb{R})

(1.7)

holds with some $p\geq 3$ for $j=0,1,2$ .

A typical example of our nonlinearity is

\displaystyle\partial_{x}\left(N(\partial_{x}u)\right)=\partial_{x}(\partial_{x}u)^{2}+\partial_{x}\left(|\partial_{x}u|^{p-1}\partial_{x}u\right)

(1.8)

with $p\geq 3$ .

Remark 1.1.

The assumption (A) implies

\displaystyle\frac{-\beta+1}{\alpha-\beta+1}<2<p.

(1.9)

This means that the nonlinearity is supercritical (see the argument in Section 4.1).

We further prepare the following notations. Let $G=G(t,x)$ be the Gaussian, that is,

\displaystyle G(t,x)=\frac{1}{\sqrt{4\pi t}}\exp\left(-\frac{x^{2}}{4t}\right).

(1.10)

We define

\displaystyle r(t)=\frac{a(t)}{b(t)}\quad\text{and}\quad R(t)=\int_{0}^{t}r(\tau)\,d\tau.

(1.11)

Remark that

\displaystyle C^{-1}(1+t)^{\alpha-\beta+1}\leq R(t)\leq C(1+t)^{\alpha-\beta+1}

(1.12)

holds with some constant $C\geq 1$ , thanks to the assumption (A).

Theorem 1.2.

Under the assumptions (A) and (N), there exists a constant $\varepsilon_{0}>0$ such that if $(u_{0},u_{1})\in\left(H^{2,1}(\mathbb{R})\cap H^{3,0}(\mathbb{R})\right)\times\left(H^{0,1}(\mathbb{R})\cap H^{1,0}(\mathbb{R})\right)$ and

\displaystyle\|u_{0}\|_{H^{2,1}\cap H^{3,0}}+\|u_{1}\|_{H^{0,1}\cap H^{1,0}}\leq\varepsilon_{0},

(1.13)

then there exists a unique solution

\displaystyle u\in C([0,\infty);H^{2,1}(\mathbb{R})\cap H^{3,0}(\mathbb{R}))\cap C^{1}([0,\infty);H^{0,1}(\mathbb{R})\cap H^{1,0}(\mathbb{R})).

(1.14)

Moreover, the solution $u$ has the asymptotic behavior

\displaystyle\|u(t,\cdot)-m^{*}G(R(t),\cdot)\|_{L^{2}}\leq C(R(t)+1)^{-\frac{1}{4}-\frac{\lambda}{2}}\left(\|u_{0}\|_{H^{2,1}\cap H^{3,0}}+\|u_{1}\|_{H^{0,1}\cap H^{1,0}}\right)

(1.15)

with some constants $C>0$ , $m^{*}\in\mathbb{R}$ , and $\lambda>0$ .

Remark 1.3.

From the proof, $\lambda$ is taken to be arbitrary so that

0<\lambda<\min\left\{\frac{1}{2},\frac{2(\beta+1)}{\alpha-\beta+1},\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right\}.

The proof is based on the method by Gallay and Raugel [5] using the self-similar transformation and the standard energy method.

This paper is organized as follows. In Section 2, we rewrite the problem through the self-similar transformation. Thereafter, we show several energy estimates in Section 3, and give a priori estimates through the estimates for the nonlinear terms in Section 4. In the appendix, we also give a lemma for energy identities which is frequently used in the proof and the proof of local-in-time existence of solution for the readers’ convenience.

At the end of this section we prepare notation and several definitions used throughout this paper. We denote by $C$ a positive constant, which may change from line to line. The symbol $a(t)\sim b(t)$ means that $C^{-1}b(t)\leq a(t)\leq Cb(t)$ holds for some constant $C\geq 1$ . $L^{p}=L^{p}(\mathbb{R})$ stands for the usual Lebesgue space, and $H^{k,m}=H^{k,m}(\mathbb{R})$ for $k\in\mathbb{Z}_{\geq 0}$ and $m\in\mathbb{R}$ is the weighted Sobolev space defined by

\displaystyle H^{k,m}(\mathbb{R})=\left\{f\in L^{2}(\mathbb{R});\|f\|_{H^{k,m}}=\sum_{\ell=0}^{k}\|(1+|x|)^{m}\partial_{x}^{\ell}f\|_{L^{2}}<\infty\right\}.

2. Scaling variables

The local existence of the solution in the class (1.14) is standard (see Appendix B). Thus, it suffices to show the a priori estimate. To prove it, following the argument of Gallay and Raugel [5], we introduce the scaling variables

\displaystyle s=\log(R(t)+1),\quad y=\frac{x}{\sqrt{R(t)+1}},

(2.1)

and define $v=v(s,y)$ and $w=w(s,y)$ by

	$\displaystyle u(t,x)$	$\displaystyle=\frac{1}{\sqrt{R(t)+1}}v\left(\log(R(t)+1),\frac{x}{\sqrt{R(t)+1}}\right),$		(2.2)
	$\displaystyle u_{t}(t,x)$	$\displaystyle=\frac{R^{\prime}(t)}{(R(t)+1)^{3/2}}w\left(\log(R(t)+1),\frac{x}{\sqrt{R(t)+1}}\right).$		(2.3)

Then, by a straightforward computation, the Cauchy problem (1.1) is rewritten as

\displaystyle\left\{\begin{aligned} &v_{s}-\frac{y}{2}v_{y}-\frac{1}{2}v=w,\\ &\frac{r^{2}e^{-s}}{a}\left(w_{s}-\frac{y}{2}w_{y}-\frac{3}{2}w\right)+\left(1+\frac{r^{\prime}}{a}\right)w=v_{yy}-\frac{e^{-s}}{a}v_{yyyy}+\frac{e^{s}}{a}\partial_{y}\left(N\left(e^{-s}v_{y}\right)\right),\\ &v(0,y)=v_{0}(y),\ w(0,y)=w_{0}(y),\end{aligned}\right.

(2.4)

where $v_{0}(y)=u_{0}(y)$ and $w_{0}(y)=\frac{1}{r(0)}u_{1}(y)$ . Here, we also note that the functions $a,b,r,r^{\prime}$ appearing the above precisely mean such as $a(t(s))=a(R^{-1}(e^{s}-1))$ .

Remark 2.1.

When $N(z)=|z|^{p-1}z$ , the nonlinearity has a bound

\displaystyle\left|\frac{e^{s}}{a(t(s))}\partial_{y}\left(N(e^{-s}v_{y})\right)\right|\leq Ce^{\left(\frac{-\beta+1}{\alpha-\beta+1}-p\right)s}|v_{y}|^{p-1}|v_{yy}|.

(2.5)

Since $\frac{-\beta+1}{\alpha-\beta+1}<2$ , the assumption (N) implies that the nonlinearity can be treated as remainder.

To investigate the asymptotic behavior of the solution of (2.4), we define

\displaystyle m(s):=\int_{\mathbb{R}}v(s,y)\,dy.

(2.6)

By the first equation of (2.4) and the integration by parts, we have

\displaystyle m_{s}(s)=\frac{d}{ds}m(s)=\int_{\mathbb{R}}v_{s}\,dy=\int_{\mathbb{R}}\left(\frac{y}{2}v_{y}+\frac{1}{2}v+w\right)\,dy=\int_{\mathbb{R}}w\,dy.

(2.7)

We also define

	$\displaystyle\phi(y)$	$\displaystyle:=G(1,y)=\frac{1}{\sqrt{4\pi}}\exp\left(-\frac{y^{2}}{4}\right),$		(2.8)
	$\displaystyle\psi(y)$	$\displaystyle:=\phi_{yy}(y).$		(2.9)

Using them, we decompose $(v,w)$ as

\displaystyle\begin{aligned} v(s,y)&=m(s)\phi(y)+f(s,y),\\ w(s,y)&=m_{s}(s)\phi(y)+m(s)\psi(y)+g(s,y),\end{aligned}

(2.10)

and we expect that the functions $f$ and $g$ defined above can be treated as remainders.

By a direct calculation, we obtain the following equation for $m(s)$ :

Lemma 2.2.

We have

\displaystyle\frac{r^{2}e^{-s}}{a}\left(m_{ss}-m_{s}\right)=-\left(1+\frac{r^{\prime}}{a}\right)m_{s}.

(2.11)

From the above lemma and the straightforward computation, we can see that $(f,g)$ satisfies the following equations.

\displaystyle\left\{\begin{aligned} &f_{s}-\frac{y}{2}f_{y}-\frac{1}{2}f=g,\\ &\frac{r^{2}e^{-s}}{a}\left(g_{s}-\frac{y}{2}g_{y}-\frac{3}{2}g\right)+\left(1+\frac{r^{\prime}}{a}\right)g=f_{yy}-\frac{e^{-s}}{a}f_{yyyy}+\frac{e^{s}}{a}\partial_{y}\left(N\left(e^{-s}v_{y}\right)\right)+h,\end{aligned}\right.

(2.12)

where

\displaystyle h

\displaystyle=-\frac{r^{2}e^{-s}}{a}\left(2m_{s}\psi-\frac{y}{2}m\psi_{y}-\frac{3}{2}m\psi\right)-\frac{r^{\prime}}{a}m\psi-\frac{e^{-s}}{a}m\psi_{yy}.

(2.13)

They satisfy

\displaystyle\int_{\mathbb{R}}f(s,y)\,dy=\int_{\mathbb{R}}g(s,y)\,dy=\int_{\mathbb{R}}h(s,y)\,dy=0.

(2.14)

Therefore, our goal is to give energy estimates of the solutions $(f,g)$ to the equation (2.12) under the condition (2.14).

3. Energy estimates

In this section, we give energy estimates of $(f,g)$ defined by (2.10). We first prepare the following general lemma for energy identities.

Lemma 3.1.

Let $l,m\in\mathbb{R}$ , $n\in\mathbb{N}\cup\{0\}$ , and let $c_{1}(s),c_{2}(s),c_{4}(s)$ be smooth functions defined on $[0,\infty)$ . We consider a system for two functions $f=f(s,y)$ and $g=g(s,y)$ given by

\displaystyle\left\{\begin{aligned} &f_{s}-\frac{y}{2}f_{y}-lf=g,\\ &c_{1}(s)\left(g_{s}-\frac{y}{2}g_{y}-mg\right)+c_{2}(s)g+g=f_{yy}-c_{4}(s)f_{yyyy}+h\end{aligned}\right.\quad(s,y)\in(0,\infty)\times\mathbb{R},

(3.1)

where $h=h(s,y)$ is a given smooth function belonging to $C([0,\infty);H^{0,n}(\mathbb{R}))$ . We define the energies

	$\displaystyle E_{1}(s)$	$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}y^{2n}\left(f_{y}^{2}+c_{4}(s)f_{yy}^{2}+c_{1}(s)g^{2}\right)\,dy,$		(3.2)
	$\displaystyle E_{2}(s)$	$\displaystyle=\int_{\mathbb{R}}y^{2n}\left(\frac{1}{2}f^{2}+c_{1}(s)fg\right)\,dy.$		(3.3)

Then, we have

$\displaystyle\frac{d}{ds}E_{1}(s)$	$\displaystyle=-\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\left(-\frac{2n-1}{4}+l\right)\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy+\left(-\frac{2n-3}{4}+l\right)c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy$	(3.4)
	$\displaystyle\quad+\left(-\frac{2n+1}{4}+m\right)c_{1}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(3.5)
	$\displaystyle\quad-2n\int_{\mathbb{R}}y^{2n-1}f_{y}g\,dy-2n(2n-1)c_{4}(s)\int_{\mathbb{R}}y^{2n-2}f_{yy}g\,dy-4nc_{4}(s)\int_{\mathbb{R}}y^{2n-1}f_{yy}g_{y}\,dy$	(3.6)
	$\displaystyle\quad+\frac{c_{4}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+\frac{c_{1}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\int_{\mathbb{R}}y^{2n}gh\,dy$	(3.7)

and

$\displaystyle\frac{d}{ds}E_{2}(s)$	$\displaystyle=-\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy-c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+\left(-\frac{2n+1}{4}+l\right)\int_{\mathbb{R}}y^{2n}f^{2}\,dy$	(3.8)
	$\displaystyle\quad+c_{1}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\left(-\frac{2n+1}{2}+l+m\right)c_{1}(s)\int_{\mathbb{R}}y^{2n}fg\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}fg\,dy$	(3.9)
	$\displaystyle\quad-2n\int_{\mathbb{R}}y^{2n-1}ff_{y}\,dy-4nc_{4}(s)\int_{\mathbb{R}}y^{2n-1}f_{y}f_{yy}\,dy-2n(2n-1)c_{4}(s)\int_{\mathbb{R}}y^{2n-2}ff_{yy}\,dy$	(3.10)
	$\displaystyle\quad+c_{1}^{\prime}(s)\int_{\mathbb{R}}y^{2n}fg\,dy+\int_{\mathbb{R}}y^{2n}fh\,dy.$	(3.11)

The case $n=0$ is given by [17, Lemma 3.1]. We will prove a slightly more general version of this lemma in Appendix A.

To bring out the decay property of the solutions $(f,g)$ to (2.12) from the condition (2.14), we define the auxiliary functions

\displaystyle F(s,y):=\int_{-\infty}^{y}f(s,z)\,dz,\quad G(s,y):=\int_{-\infty}^{y}g(s,z)\,dz,\quad H(s,y):=\int_{-\infty}^{y}h(s,z)\,dz.

(3.12)

Then, by the following Hardy inequality, the conditions (2.14) and $f(s),g(s)\in H^{0,1}(\mathbb{R})$ ensure $F(s),G(s)\in L^{2}(\mathbb{R})$ .

Lemma 3.2 (Hardy-type inequality [13, Lemma 3.9]).

Let $f=f(y)\in H^{0,1}(\mathbb{R})$ and satisfy $\int_{\mathbb{R}}f(y)\,dy=0$ . Let $F(y)=\int_{-\infty}^{y}f(z)\,dz$ . Then, we have

\displaystyle\int_{\mathbb{R}}F(y)^{2}\,dy\leq 4\int_{\mathbb{R}}y^{2}f(y)^{2}\,dy.

(3.13)

From (2.12), $F$ and $G$ satisfy the following system.

\displaystyle\left\{\begin{aligned} &F_{s}-\frac{y}{2}F_{y}=G,\\ &\frac{r^{2}e^{-s}}{a}\left(G_{s}-\frac{y}{2}G_{y}-G\right)+\left(1+\frac{r^{\prime}}{a}\right)G=F_{yy}-\frac{e^{-s}}{a}F_{yyyy}+\frac{e^{s}}{a}N\left(e^{-s}v_{y}\right)+H.\end{aligned}\right.

(3.14)

We define the energies of $(F,G)$ by

	$\displaystyle E_{01}(s)$	$\displaystyle:=\frac{1}{2}\int_{\mathbb{R}}F_{y}(s,y)^{2}\,dy+\frac{e^{-s}}{2a}\int_{\mathbb{R}}F_{yy}(s,y)^{2}\,dy+\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}G(s,y)^{2}\,dy,$		(3.15)
	$\displaystyle E_{02}(s)$	$\displaystyle:=\frac{1}{2}\int_{\mathbb{R}}F(s,y)^{2}\,dy+\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}F(s,y)G(s,y)\,dy.$		(3.16)

Lemma 3.3.

We have

$\displaystyle\frac{d}{ds}E_{01}(s)+\int_{\mathbb{R}}G^{2}\,dy$	$\displaystyle=\frac{1}{2}E_{01}(s)-\frac{a^{\prime}}{2ra^{2}}\int_{\mathbb{R}}F_{yy}^{2}\,dy-\frac{ra^{\prime}}{2a^{2}}\int_{\mathbb{R}}G^{2}\,dy$	(3.17)
	$\displaystyle\quad+\int_{\mathbb{R}}G\frac{e^{s}}{a}N\left(e^{-s}v_{y}\right)\,dy+\int_{\mathbb{R}}GH\,dy,$	(3.18)
$\displaystyle\frac{d}{ds}E_{02}(s)+\frac{1}{2}E_{02}(s)+2E_{01}(s)$	$\displaystyle=2\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}G^{2}\,dy+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}FG\,dy$	(3.19)
	$\displaystyle\quad+\int_{\mathbb{R}}F\frac{e^{s}}{a}N\left(e^{-s}v_{y}\right)\,dy+\int_{\mathbb{R}}FH\,dy.$	(3.20)

Proof.

We apply Lemma 3.1 as $f=F$ , $g=G$ and $h=\frac{e^{s}}{a}N(e^{-s}v_{y})+H$ with $l=0$ , $m=1$ , $n=0$ , $c_{1}(s)=r^{2}e^{-s}/a$ , $c_{2}(s)=r^{\prime}/a$ , and $c_{4}(s)=e^{-s}/a$ . Noting that

\displaystyle\frac{d}{ds}r(t(s))=\frac{d}{ds}r(R^{-1}(e^{s}-1))=\frac{r^{\prime}(t(s))}{r(t(s))}e^{s},

(3.21)

we first have

	$\displaystyle c_{1}^{\prime}(s)$	$\displaystyle=\frac{1}{a^{2}}\left(2r\frac{dr}{ds}ae^{-s}-r^{2}ae^{-s}-r^{2}\frac{da}{ds}e^{-s}\right)=\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}},$		(3.22)
	$\displaystyle c_{4}^{\prime}(s)$	$\displaystyle=\frac{1}{a^{2}}\left(-ae^{-s}-\frac{da}{ds}e^{-s}\right)=-\frac{e^{-s}}{a}-\frac{a^{\prime}}{ra^{2}}.$		(3.23)

Thus, we obtain

$\displaystyle\frac{d}{ds}E_{01}(s)$	$\displaystyle=-\int_{\mathbb{R}}G^{2}\,dy+\frac{1}{4}\int_{\mathbb{R}}F_{y}^{2}\,dy+\frac{3e^{-s}}{4a}\int_{\mathbb{R}}F_{yy}^{2}\,dy$	(3.24)
	$\displaystyle\quad+\frac{3r^{2}e^{-s}}{4a}\int_{\mathbb{R}}G^{2}\,dy-\frac{r^{\prime}}{a}\int_{\mathbb{R}}G^{2}\,dy-\frac{1}{2}\left(\frac{e^{-s}}{a}+\frac{a^{\prime}}{ra^{2}}\right)\int_{\mathbb{R}}F_{yy}^{2}\,dy$	(3.25)
	$\displaystyle\quad+\frac{1}{2}\left(\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}G^{2}\,dy$	(3.26)
	$\displaystyle\quad+\int_{\mathbb{R}}G\frac{e^{s}}{a}N\left(e^{-s}v_{y}\right)\,dy+\int_{\mathbb{R}}GH\,dy$	(3.27)
	$\displaystyle=-\int_{\mathbb{R}}G^{2}\,dy+\frac{1}{2}E_{01}(s)-\frac{a^{\prime}}{2ra^{2}}\int_{\mathbb{R}}F_{yy}^{2}\,dy-\frac{ra^{\prime}}{2a^{2}}\int_{\mathbb{R}}G^{2}\,dy$	(3.28)
	$\displaystyle\quad+\int_{\mathbb{R}}G\frac{e^{s}}{a}N\left(e^{-s}v_{y}\right)\,dy+\int_{\mathbb{R}}GH\,dy.$	(3.29)

Next, we compute the derivative of $E_{02}(s)$ . By Lemma 3.1, we obtain

$\displaystyle\frac{d}{ds}E_{02}(s)$	$\displaystyle=-\int_{\mathbb{R}}F_{y}^{2}\,dy-\frac{e^{-s}}{a}\int_{\mathbb{R}}F_{yy}^{2}\,dy-\frac{1}{4}\int_{\mathbb{R}}F^{2}\,dy$	(3.30)
	$\displaystyle\quad+\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}G^{2}\,dy+\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}FG\,dy-\frac{r^{\prime}}{a}\int_{\mathbb{R}}FG\,dy$	(3.31)
	$\displaystyle\quad+\int_{\mathbb{R}}F\frac{e^{s}}{a}N\left(e^{-s}v_{y}\right)\,dy+\int_{\mathbb{R}}FH\,dy$	(3.32)
	$\displaystyle\quad+\left(\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}FG\,dy$	(3.33)
	$\displaystyle=-\frac{1}{2}E_{02}(s)-2E_{01}(s)$	(3.34)
	$\displaystyle\quad+2\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}G^{2}\,dy+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}FG\,dy$	(3.35)
	$\displaystyle\quad+\int_{\mathbb{R}}F\frac{e^{s}}{a}N\left(e^{-s}v_{y}\right)\,dy+\int_{\mathbb{R}}FH\,dy.$	(3.36)

This completes the proof. ∎

Next, for $n=0,1$ , we define the energies of $(f,g)$ by

	$\displaystyle E_{11}^{(n)}(s)$	$\displaystyle:=\frac{1}{2}\int_{\mathbb{R}}y^{2n}f_{y}(s,y)^{2}\,dy+\frac{e^{-s}}{2a}\int_{\mathbb{R}}y^{2n}f_{yy}(s,y)^{2}\,dy+\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}y^{2n}g(s,y)^{2}\,dy,$		(3.37)
	$\displaystyle E_{12}^{(n)}(s)$	$\displaystyle:=\frac{1}{2}\int_{\mathbb{R}}y^{2n}f(s,y)^{2}\,dy+\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}y^{2n}f(s,y)g(s,y)\,dy.$		(3.38)

Lemma 3.4.

For $n=0,1$ , we have

$\displaystyle\frac{d}{ds}E_{11}^{(n)}(s)+\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	$\displaystyle=\frac{3-2n}{2}E_{11}^{(n)}(s)-2n\int_{\mathbb{R}}y^{2n-1}f_{y}g\,dy$	(3.39)
	$\displaystyle\quad-2n(2n-1)\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-2}f_{yy}g\,dy-4n\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-1}f_{yy}g_{y}\,dy$	(3.40)
	$\displaystyle\quad-\frac{a^{\prime}}{2ra^{2}}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy-\frac{ra^{\prime}}{2a^{2}}\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(3.41)
	$\displaystyle\quad+\int_{\mathbb{R}}y^{2n}g\left(\frac{e^{s}}{a}\partial_{y}(N(e^{-s}v_{y}))+h\right)\,dy$	(3.42)

and

$\displaystyle\frac{d}{ds}E_{12}^{(n)}(s)+\frac{1}{2}E_{12}^{(n)}(s)+2E_{11}^{(n)}(s)$	$\displaystyle=(1-n)E_{12}^{(n)}(s)+\frac{2r^{2}e^{-s}}{a}\int_{\mathbb{R}}y^{2n}g^{2}\,dy-2n\int_{\mathbb{R}}y^{2n-1}ff_{y}\,dy$	(3.43)
	$\displaystyle\quad-4n\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-1}f_{y}f_{yy}\,dy-2n(2n-1)\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-2}ff_{yy}\,dy$	(3.44)
	$\displaystyle\quad+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}y^{2n}fg\,dy+\int_{\mathbb{R}}y^{2n}f\left(\frac{e^{s}}{a}\partial_{y}(N(e^{-s}v_{y}))+h\right)\,dy.$	(3.45)

Proof.

For $n=0,1$ , we apply Lemma 3.1 as $f=f$ , $g=g$ and $h=\frac{e^{s}}{a}\partial_{y}N(e^{-s}v_{y})+h$ with $l=\frac{1}{2}$ , $m=\frac{3}{2}$ , $c_{1}(s)=r^{2}e^{-s}/a$ , $c_{2}(s)=r^{\prime}/a$ , and $c_{4}(s)=e^{-s}/a$ . Using (3.22) and (3.23), we have

$\displaystyle\frac{d}{ds}E_{11}^{(n)}(s)$	$\displaystyle=-\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\frac{3-2n}{4}\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy+\frac{5-2n}{4}\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy$	(3.46)
	$\displaystyle\quad+\frac{5-2n}{4}\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}y^{2n}g^{2}\,dy-\frac{r^{\prime}}{a}\int_{\mathbb{R}}y^{2n}g^{2}\,dy-2n\int_{\mathbb{R}}y^{2n-1}f_{y}g\,dy$	(3.47)
	$\displaystyle\quad-2n(2n-1)\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-2}f_{yy}g\,dy-4n\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-1}f_{yy}g_{y}\,dy$	(3.48)
	$\displaystyle\quad-\frac{1}{2}\left(\frac{e^{-s}}{a}+\frac{a^{\prime}}{ra^{2}}\right)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+\frac{1}{2}\left(\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(3.49)
	$\displaystyle\quad+\int_{\mathbb{R}}y^{2n}g\left(\frac{e^{s}}{a}\partial_{y}(N(e^{-s}v_{y}))+h\right)\,dy$	(3.50)
	$\displaystyle=-\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\frac{3-2n}{2}E_{11}^{(n)}(s)$	(3.51)
	$\displaystyle\quad-2n\int_{\mathbb{R}}y^{2n-1}f_{y}g\,dy-2n(2n-1)\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-2}f_{yy}g\,dy-4n\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-1}f_{yy}g_{y}\,dy$	(3.52)
	$\displaystyle\quad-\frac{a^{\prime}}{2ra^{2}}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy-\frac{ra^{\prime}}{2a^{2}}\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(3.53)
	$\displaystyle\quad+\int_{\mathbb{R}}y^{2n}g\left(\frac{e^{s}}{a}\partial_{y}(N(e^{-s}v_{y}))+h\right)\,dy.$	(3.54)

Similarly, we calculate

$\displaystyle\frac{d}{ds}E_{12}^{(n)}(s)$	$\displaystyle=-\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy-\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+\frac{1-2n}{4}\int_{\mathbb{R}}y^{2n}f^{2}\,dy$	(3.55)
	$\displaystyle\quad+\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\frac{3-2n}{2}\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}y^{2n}fg\,dy-\frac{r^{\prime}}{a}\int_{\mathbb{R}}y^{2n}fg\,dy$	(3.56)
	$\displaystyle\quad-2n\int_{\mathbb{R}}y^{2n-1}ff_{y}\,dy-4n\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-1}f_{y}f_{yy}\,dy-2n(2n-1)\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-2}ff_{yy}\,dy$	(3.57)
	$\displaystyle\quad+\left(\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}y^{2n}fg\,dy+\int_{\mathbb{R}}y^{2n}f\left(\frac{e^{s}}{a}\partial_{y}(N(e^{-s}v_{y}))+h\right)\,dy$	(3.58)
	$\displaystyle=\frac{1-2n}{2}E_{12}^{(n)}(s)-2E_{11}^{(n)}(s)+\frac{2r^{2}e^{-s}}{a}\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(3.59)
	$\displaystyle\quad-2n\int_{\mathbb{R}}y^{2n-1}ff_{y}\,dy-4n\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-1}f_{y}f_{yy}\,dy-2n(2n-1)\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2n-2}ff_{yy}\,dy$	(3.60)
	$\displaystyle\quad+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}y^{2n}fg\,dy+\int_{\mathbb{R}}y^{2n}f\left(\frac{e^{s}}{a}\partial_{y}(N(e^{-s}v_{y}))+h\right)\,dy.$	(3.61)

This completes the proof. ∎

Finally, to control the bad term $\displaystyle-4\frac{e^{-s}}{a}\int_{\mathbb{R}}yf_{yy}g_{y}\,dy$ in $\dfrac{d}{ds}E_{11}^{(1)}(s)$ , we consider the energies

	$\displaystyle E_{21}(s)$	$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}\left(f_{yy}^{2}+\frac{e^{-s}}{a}f_{yyy}^{2}+\frac{r^{2}e^{-s}}{a}g_{y}^{2}\right)\,dy,$		(3.62)
	$\displaystyle E_{22}(s)$	$\displaystyle=\int_{\mathbb{R}}\left(\frac{1}{2}f_{y}^{2}+\frac{r^{2}e^{-s}}{a}f_{y}g_{y}\right)\,dy.$		(3.63)

Since $(f_{y},g_{y})$ satisfies the equations

\displaystyle\left\{\begin{aligned} &(f_{y})_{s}-\frac{y}{2}(f_{y})_{y}-f_{y}=g_{y},\\ &\frac{r^{2}e^{-s}}{a}\left((g_{y})_{s}-\frac{y}{2}(g_{y})_{y}-2g_{y}\right)+g_{y}+\frac{r^{\prime}}{a}g_{y}=(f_{y})_{yy}-\frac{e^{-s}}{a}(f_{y})_{yyyy}+\frac{e^{s}}{a}\partial_{y}^{2}N(e^{-s}v_{y}))+h_{y},\end{aligned}\right.

(3.64)

we have the following lemma.

Lemma 3.5.

We have

	$\displaystyle\frac{d}{ds}E_{21}(s)+\int_{\mathbb{R}}g_{y}^{2}\,dy$	$\displaystyle=\frac{5}{2}E_{21}(s)-\frac{a^{\prime}}{2ra^{2}}\int_{\mathbb{R}}f_{yyy}^{2}\,dy-\frac{ra^{\prime}}{2a^{2}}\int_{\mathbb{R}}g_{y}^{2}\,dy$		(3.65)
		$\displaystyle\quad+\int_{\mathbb{R}}g_{y}\left(\frac{e^{s}}{a}\partial_{y}^{2}(N(e^{-s}v_{y}))+h_{y}\right)\,dy$		(3.66)

and

	$\displaystyle\frac{d}{ds}E_{22}(s)+2E_{21}(s)$	$\displaystyle=\frac{3}{2}E_{22}(s)+2\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}g_{y}^{2}\,dy+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}f_{y}g_{y}\,dy$		(3.67)
		$\displaystyle\quad+\int_{\mathbb{R}}f_{y}\left(\frac{e^{s}}{a}\partial_{y}^{2}(N(e^{-s}v_{y}))+h_{y}\right)\,dy.$		(3.68)

Proof.

Applying Lemma 3.1 as $f=f_{y}$ , $g=g_{y}$ and $h=\frac{e^{s}}{a}\partial_{yy}N(e^{-s}v_{y})+h_{y}$ with $l=1$ , $m=2$ , $n=0$ , $c_{1}(s)=\frac{r^{2}e^{-s}}{a}$ , $c_{2}(s)=\frac{r^{\prime}}{a}$ , and $c_{4}(s)=\frac{e^{-s}}{a}$ , and also using (3.22) and (3.23), we have

$\displaystyle\frac{d}{ds}E_{21}(s)$	$\displaystyle=-\int_{\mathbb{R}}g_{y}^{2}\,dy+\frac{5}{4}\int_{\mathbb{R}}f_{yy}^{2}\,dy+\frac{7}{4}\frac{e^{-s}}{a}\int_{\mathbb{R}}f_{yyy}^{2}\,dy$	(3.69)
	$\displaystyle\quad+\frac{7}{4}\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}g_{y}^{2}\,dy-\frac{r^{\prime}}{a}\int_{\mathbb{R}}g_{y}^{2}\,dy$	(3.70)
	$\displaystyle\quad-\frac{1}{2}\left(\frac{e^{-s}}{a}+\frac{a^{\prime}}{ra^{2}}\right)\int_{\mathbb{R}}f_{yyy}^{2}\,dy+\frac{1}{2}\left(\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}g_{y}^{2}\,dy$	(3.71)
	$\displaystyle\quad+\int_{\mathbb{R}}g_{y}\left(\frac{e^{s}}{a}\partial_{y}^{2}(N(e^{-s}v_{y}))+h_{y}\right)\,dy$	(3.72)
	$\displaystyle=-\int_{\mathbb{R}}g_{y}^{2}\,dy+\frac{5}{2}E_{21}(s)$	(3.73)
	$\displaystyle\quad-\frac{a^{\prime}}{2ra^{2}}\int_{\mathbb{R}}f_{yyy}^{2}\,dy-\frac{ra^{\prime}}{2a^{2}}\int_{\mathbb{R}}g_{y}^{2}\,dy$	(3.74)
	$\displaystyle\quad+\int_{\mathbb{R}}g_{y}\left(\frac{e^{s}}{a}\partial_{y}^{2}(N(e^{-s}v_{y}))+h_{y}\right)\,dy.$	(3.75)

Similarly, we have

$\displaystyle\frac{d}{ds}E_{22}(s)$	$\displaystyle=-\int_{\mathbb{R}}f_{yy}^{2}\,dy-\frac{e^{-s}}{a}\int_{\mathbb{R}}f_{yyy}^{2}\,dy+\frac{3}{4}\int_{\mathbb{R}}f_{y}^{2}\,dy$	(3.76)
	$\displaystyle\quad+\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}g_{y}^{2}\,dy+\frac{5}{2}\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}f_{y}g_{y}\,dy-\frac{r^{\prime}}{a}\int_{\mathbb{R}}f_{y}g_{y}\,dy$	(3.77)
	$\displaystyle\quad+\left(\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}f_{y}g_{y}\,dy+\int_{\mathbb{R}}f_{y}\left(\frac{e^{s}}{a}\partial_{y}^{2}(N(e^{-s}v_{y}))+h_{y}\right)\,dy$	(3.78)
	$\displaystyle=\frac{3}{2}E_{22}(s)-2E_{21}(s)$	(3.79)
	$\displaystyle\quad+2\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}g_{y}^{2}\,dy+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}f_{y}g_{y}\,dy$	(3.80)
	$\displaystyle\quad+\int_{\mathbb{R}}f_{y}\left(\frac{e^{s}}{a}\partial_{y}^{2}(N(e^{-s}v_{y}))+h_{y}\right)\,dy.$	(3.81)

This completes the proof. ∎

Finally, we define

	$\displaystyle E_{m1}(s)$	$\displaystyle:=\frac{1}{2}\frac{r^{2}e^{-s}}{a}m_{s}(s)^{2},$		(3.82)
	$\displaystyle E_{m2}(s)$	$\displaystyle:=\frac{1}{2}m(s)^{2}+\frac{r^{2}e^{-s}}{a}m(s)m_{s}(s).$		(3.83)

Then, we have the following energy identities.

Lemma 3.6.

We have

\displaystyle\frac{d}{ds}E_{m1}(s)+\frac{1}{2}E_{m1}(s)+m_{s}(s)^{2}

\displaystyle=\left(\frac{3r^{2}e^{-s}}{4a}-\frac{ra^{\prime}}{2a^{2}}\right)m_{s}(s)^{2}

(3.84)

and

\displaystyle\frac{d}{ds}E_{m2}(s)

\displaystyle=2E_{m1}(s)+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)m(s)m_{s}(s).

(3.85)

Proof.

By (3.22) and Lemma 2.2, we have

$\displaystyle\frac{d}{ds}E_{m1}(s)$	$\displaystyle=\frac{1}{2}\frac{d}{ds}\left(\frac{r^{2}e^{-s}}{a}\right)m_{s}^{2}+\frac{r^{2}e^{-s}}{a}m_{s}m_{ss}$	(3.86)
	$\displaystyle=\frac{1}{2}\left(\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}}\right)m_{s}^{2}$	(3.87)
	$\displaystyle\quad+\frac{r^{2}e^{-s}}{a}m_{s}^{2}-\left(1+\frac{r^{\prime}}{a}\right)m_{s}^{2}$	(3.88)
	$\displaystyle=\frac{1}{2}\frac{r^{2}e^{-s}}{a}m_{s}^{2}-\frac{1}{2}\frac{ra^{\prime}}{a^{2}}m_{s}^{2}-m_{s}^{2}$	(3.89)
	$\displaystyle=-\frac{1}{2}E_{m1}(s)-m_{s}^{2}+\left(\frac{3r^{2}e^{-s}}{4a}-\frac{ra^{\prime}}{2a^{2}}\right)m_{s}^{2}.$	(3.90)

Similarly, we have

$\displaystyle\frac{d}{ds}E_{m2}(s)$	$\displaystyle=mm_{s}+\frac{d}{ds}\left(\frac{r^{2}e^{-s}}{a}\right)mm_{s}+\frac{r^{2}e^{-s}}{a}m_{s}^{2}+\frac{r^{2}e^{-s}}{a}mm_{ss}$	(3.91)
	$\displaystyle=mm_{s}+\left(\frac{2r^{\prime}}{a}-\frac{r^{2}e^{-s}}{a}-\frac{ra^{\prime}}{a^{2}}\right)mm_{s}$	(3.92)
	$\displaystyle\quad+\frac{r^{2}e^{-s}}{a}m_{s}^{2}+\frac{r^{2}e^{-s}}{a}mm_{s}-\left(1+\frac{r^{\prime}}{a}\right)mm_{s}$	(3.93)
	$\displaystyle=\frac{r^{2}e^{-s}}{a}m_{s}^{2}+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)mm_{s}$	(3.94)
	$\displaystyle=2E_{m1}(s)+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)mm_{s}.$	(3.95)

∎

3.1. Energy estimates

Now, we combine the energy identities in the previous subsection to obtain the energy estimates. First, we prepare the following estimates for remainders.

Lemma 3.7.

Set

\displaystyle\delta

\displaystyle:=\min\left\{\frac{\beta+1}{\alpha-\beta+1},\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right\},

(3.96)

which is positive if $(\alpha,\beta)\in\Omega_{1}$ . Then, we have

\displaystyle\frac{r^{2}e^{-s}}{a}\sim e^{-\frac{\beta+1}{\alpha-\beta+1}s}\leq Ce^{-\delta s},\quad\frac{e^{-s}}{a}\sim e^{-\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\leq Ce^{-\delta s},

(3.97)

and

\displaystyle\left|\frac{a^{\prime}}{ra^{2}}\right|\leq C\frac{e^{-s}}{a},\quad\left|\frac{ra^{\prime}}{a^{2}}\right|\leq C\frac{r^{2}e^{-s}}{a},\quad\left|\frac{r^{\prime}}{a}\right|\leq C\frac{r^{2}e^{-s}}{a}.

(3.98)

Define

\displaystyle\mathbb{E}_{0}(s)

\displaystyle:=E_{01}(s)+c_{0}E_{02}(s),

(3.99)

where $c_{0}>0$ is a sufficiently large constant determined later.

Lemma 3.8.

There exists a constant $c_{0}>0$ satisfying the following: For any $\eta>0$ , there exists $s_{0}>0$ such that for any $s\geq s_{0}$ , we have

\displaystyle\mathbb{E}_{0}(s)\geq C\left(\int_{\mathbb{R}}F_{y}^{2}\,dy+\frac{e^{-s}}{2a}\int_{\mathbb{R}}F_{yy}^{2}\,dy+\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}G^{2}\,dy+\int_{\mathbb{R}}F^{2}\,dy\right)

(3.100)

and

	$\displaystyle\frac{d}{ds}\mathbb{E}_{0}(s)+\frac{1}{2}\mathbb{E}_{0}(s)+\frac{1}{4}\int_{\mathbb{R}}G^{2}\,dy$		(3.101)
	$\displaystyle\leq\eta\mathbb{E}_{0}(s)+C(\eta)\left(\\|H(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right).$		(3.102)

Proof.

Let $\eta>0$ be arbitrary. Lemmas 3.3 and 3.7, and the Schwarz inequality imply

$\displaystyle\frac{d}{ds}E_{01}(s)+\int_{\mathbb{R}}G^{2}\,dy$	$\displaystyle\leq\frac{1}{2}E_{01}(s)+C_{1}\frac{e^{-s}}{a}\int_{\mathbb{R}}F_{yy}^{2}\,dy+C_{1}\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}G^{2}\,dy$	(3.103)
	$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}}G^{2}\,dy+C\left(\\|H(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right)$	(3.104)
	$\displaystyle\leq\left(\frac{1}{2}+2C_{1}\right)E_{01}(s)+\frac{1}{2}\int_{\mathbb{R}}G^{2}\,dy$	(3.105)
	$\displaystyle\quad+C\left(\\|H(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right)$	(3.106)

with some $C_{1}>0$ and

	$\displaystyle\frac{d}{ds}E_{02}(s)+\frac{1}{2}E_{02}(s)+2E_{01}(s)$	$\displaystyle\leq C_{2}(\eta_{1})e^{-\delta s}\int_{\mathbb{R}}G^{2}\,dy+\eta_{1}\int_{\mathbb{R}}F^{2}\,dy$		(3.107)
		$\displaystyle\quad+C(\eta_{1})\left(\\|H(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right),$		(3.108)

with some $C_{2}(\eta_{1})>0$ , where $\eta_{1}$ is an arbitrary small positive number determined later. We take $c_{0}$ sufficiently large so that $2c_{0}-\frac{1}{2}-2C_{1}\geq\frac{1}{2}$ . Then, letting $s_{0}$ sufficiently large so that $c_{0}C_{2}(\eta_{1})e^{-\delta s}\leq\frac{1}{4}$ holds for any $s\geq s_{0}$ , we conclude

		$\displaystyle\frac{d}{ds}\mathbb{E}_{0}(s)+\frac{1}{2}\mathbb{E}_{0}(s)+\frac{1}{4}\int_{\mathbb{R}}G^{2}\,dy$		(3.109)
		$\displaystyle\leq 2c_{0}\eta_{1}\int_{\mathbb{R}}F^{2}\,dy+C\left(\\|H(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right).$		(3.110)

On the other hand, we remark that

	$\displaystyle\frac{r^{2}e^{-s}}{a}\left\|\int_{\mathbb{R}}F(s,y)G(s,y)\,dy\right\|$	$\displaystyle\leq\frac{1}{4}\int_{\mathbb{R}}F(s,y)^{2}\,dy+C\left(\frac{r^{2}e^{-s}}{a}\right)^{2}\int_{\mathbb{R}}G(s,y)^{2}\,dy$		(3.111)
		$\displaystyle\leq\frac{1}{4}\int_{\mathbb{R}}F(s,y)^{2}\,dy+Ce^{-\delta s}\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}G(s,y)^{2}\,dy.$		(3.112)

From this, retaking $s_{0}$ larger if needed, we have for $s\geq s_{0}$ ,

\displaystyle\mathbb{E}_{0}(s)\geq C\left(\int_{\mathbb{R}}F_{y}^{2}\,dy+\frac{e^{-s}}{2a}\int_{\mathbb{R}}F_{yy}^{2}\,dy+\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}G^{2}\,dy+\int_{\mathbb{R}}F^{2}\,dy\right),

(3.113)

which shows the first assertion. In particular, it gives $\int_{\mathbb{R}}F_{y}^{2}\,dy\leq C\mathbb{E}_{0}(s)$ Applying this to the right-hand side of (3.110) and taking $\eta_{1}$ so that $\eta=2c_{0}C\eta_{1}$ , we have the desired estimate. ∎

Next, for $n=0,1$ , we define

\displaystyle\mathbb{E}^{(n)}_{1}(s):=E^{(n)}_{11}(s)+c_{1}^{(n)}E^{(n)}_{12}(s),

(3.114)

where $c_{1}^{(0)}$ and $c_{1}^{(1)}$ are sufficiently large constants determined later. The following two lemmas are the estimates for $\mathbb{E}_{1}^{(0)}(s)$ and $\mathbb{E}_{1}^{(1)}(s)$ , respectively.

Lemma 3.9.

There exist positive constants $c_{1}^{(0)}$ and $s_{1}^{(0)}$ such that for any $s\geq s_{1}^{(0)}$ , we have

\displaystyle\mathbb{E}^{(0)}_{1}(s)\geq C\left(\int_{\mathbb{R}}f_{y}^{2}\,dy+\frac{e^{-s}}{2a}\int_{\mathbb{R}}f_{yy}^{2}\,dy+\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}g^{2}\,dy+\int_{\mathbb{R}}f^{2}\,dy\right)

(3.115)

and

	$\displaystyle\frac{d}{ds}\mathbb{E}^{(0)}_{1}(s)+\frac{1}{2}\mathbb{E}^{(0)}_{1}(s)+\frac{1}{4}\int_{\mathbb{R}}g^{2}\,dy$		(3.116)
	$\displaystyle\leq C\mathbb{E}_{0}(s)+C\left(\\|h(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\left\\|\partial_{y}N\left(e^{-s}v_{y}\right)\right\\|_{L^{2}}^{2}\right).$		(3.117)

Proof.

By Lemmas 3.4 and 3.7, and the Schwarz inequality, we have

	$\displaystyle\frac{d}{ds}E_{11}^{(0)}(s)+\frac{1}{2}E_{11}^{(0)}(s)+\int_{\mathbb{R}}g^{2}\,dy$		(3.118)
	$\displaystyle\leq 2E_{11}^{(0)}(s)+C\frac{e^{-s}}{a}\int_{\mathbb{R}}f_{yy}^{2}\,dy+C\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}g^{2}\,dy$		(3.119)
	$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}}g^{2}\,dy+C\left(\\|h(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}\left(N\left(e^{-s}v_{y}\right)\right)\\|_{L^{2}}^{2}\right),$		(3.120)

which implies

	$\displaystyle\frac{d}{ds}E_{11}^{(0)}(s)+\frac{1}{2}E_{11}^{(0)}(s)+\frac{1}{2}\int_{\mathbb{R}}g^{2}\,dy$		(3.121)
	$\displaystyle\leq\left(2+C_{1}\right)E_{11}^{(0)}(s)+C\left(\\|h(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}\left(N\left(e^{-s}v_{y}\right)\right)\\|_{L^{2}}^{2}\right)$		(3.122)

with some constant $C_{1}>0$ . In a similar way, we also obtain

	$\displaystyle\frac{d}{ds}E_{12}^{(0)}(s)+\frac{1}{2}E_{12}^{(0)}(s)+2E_{11}^{(0)}(s)$		(3.123)
	$\displaystyle\leq C\int_{\mathbb{R}}f^{2}\,dy+C_{2}e^{-\delta s}\int_{\mathbb{R}}g^{2}\,dy$		(3.124)
	$\displaystyle\quad+C\left(\\|h(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}\left(N\left(e^{-s}v_{y}\right)\right)\\|_{L^{2}}^{2}\right)$		(3.125)

with some constant $C_{2}>0$ . Therefore, taking $c_{1}^{(0)}$ and $s_{1}^{(0)}$ sufficiently large so that $2c_{1}^{(0)}-\left(2+C_{1}\right)\geq\frac{1}{2}$ and $c_{1}^{(0)}C_{2}e^{-\delta s}\leq\frac{1}{4}$ holds for any $s\geq s_{1}^{(0)}$ , we conclude

	$\displaystyle\frac{d}{ds}\mathbb{E}^{(0)}_{1}(s)+\frac{1}{2}\mathbb{E}^{(0)}_{1}(s)+\frac{1}{4}\int_{\mathbb{R}}g^{2}\,dy$		(3.126)
	$\displaystyle\leq C\int_{\mathbb{R}}f^{2}\,dy+C\left(\\|h(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\left\\|\partial_{y}N\left(e^{-s}v_{y}\right)\right\\|_{L^{2}}^{2}\right).$		(3.127)

Finally, by $\int_{\mathbb{R}}f^{2}\,dy=\int_{\mathbb{R}}F_{y}^{2}\,dy\leq C\mathbb{E}_{0}$ , the proof of the second assertion is complete. The first assertion is proved in the same way as the previous lemma and we omit the detail. ∎

Lemma 3.10.

There exists a constant $c_{1}^{(1)}>0$ satisfying the following: for any $\eta^{\prime}>0$ , there exists a constant $s_{1}^{(1)}>0$ such that for any $s\geq s_{1}^{(1)}$ , we have

\displaystyle\mathbb{E}^{(1)}_{1}(s)\geq C\left(\int_{\mathbb{R}}y^{2}f_{y}^{2}\,dy+\frac{e^{-s}}{2a}\int_{\mathbb{R}}y^{2}f_{yy}^{2}\,dy+\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}y^{2}g^{2}\,dy+\int_{\mathbb{R}}y^{2}f^{2}\,dy\right)

(3.128)

and

	$\displaystyle\frac{d}{ds}\mathbb{E}_{1}^{(1)}(s)+\frac{1}{2}\mathbb{E}_{1}^{(1)}(s)+\frac{1}{4}\int_{\mathbb{R}}y^{2}g^{2}\,dy$		(3.129)
	$\displaystyle\leq\eta^{\prime}\mathbb{E}_{1}^{(1)}(s)+C\mathbb{E}_{1}^{(0)}(s)-4\frac{e^{-s}}{a}\int_{\mathbb{R}}yf_{yy}g_{y}\,dy+Ce^{-\delta s}\int_{\mathbb{R}}g^{2}dy$		(3.130)
	$\displaystyle\quad+C(\eta^{\prime})\left(\\|yh(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|y\partial_{y}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right).$		(3.131)

Proof.

Let $\eta^{\prime}>0$ be arbitrary. By Lemmas 3.4 and 3.7, and the Schwarz inequality, we have

	$\displaystyle\frac{d}{ds}E_{11}^{(1)}(s)+\frac{1}{2}E_{11}^{(1)}(s)+\int_{\mathbb{R}}y^{2}g^{2}\,dy$		(3.132)
	$\displaystyle\leq E_{11}^{(1)}(s)+C\frac{e^{-s}}{a}\int_{\mathbb{R}}y^{2}f_{yy}^{2}\,dy+C\frac{e^{-s}}{a}\int_{\mathbb{R}}f_{yy}^{2}\,dy+C\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}y^{2}g^{2}\,dy$		(3.133)
	$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}}y^{2}g^{2}\,dy+C\int_{\mathbb{R}}f_{y}^{2}\,dy+Ce^{-\delta s}\int_{\mathbb{R}}g^{2}\,dy$		(3.134)
	$\displaystyle\quad-4\frac{e^{-s}}{a}\int_{\mathbb{R}}yf_{yy}g_{y}\,dy+C\left(\\|yh(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|y\partial_{y}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right),$		(3.135)

which implies

	$\displaystyle\frac{d}{ds}E_{11}^{(1)}(s)+\frac{1}{2}E_{11}^{(1)}(s)+\frac{1}{2}\int_{\mathbb{R}}y^{2}g^{2}\,dy$		(3.136)
	$\displaystyle\leq(1+C_{1}^{\prime})E_{11}^{(1)}(s)+C\mathbb{E}_{1}^{(0)}(s)+Ce^{-\delta s}\int_{\mathbb{R}}g^{2}\,dy$		(3.137)
	$\displaystyle\quad-4\frac{e^{-s}}{a}\int_{\mathbb{R}}yf_{yy}g_{y}\,dy+C\left(\\|yh(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|y\partial_{y}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right)$		(3.138)

with some constant $C_{1}^{\prime}>0$ . Next, for $E_{12}^{(1)}(s)$ , Lemmas 3.4 and 3.7 and the Schwarz inequality imply

	$\displaystyle\frac{d}{ds}E_{12}^{(1)}(s)+\frac{1}{2}E_{12}^{(1)}(s)+2E_{11}^{(1)}(s)$		(3.139)
	$\displaystyle=\frac{2r^{2}e^{-s}}{a}\int_{\mathbb{R}}y^{2}g^{2}\,dy-2\int_{\mathbb{R}}yff_{y}\,dy-4\frac{e^{-s}}{a}\int_{\mathbb{R}}yf_{y}f_{yy}\,dy-2\frac{e^{-s}}{a}\int_{\mathbb{R}}ff_{yy}\,dy$		(3.140)
	$\displaystyle\quad+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)\int_{\mathbb{R}}y^{2}fg\,dy+\int_{\mathbb{R}}y^{2}f\left(\frac{e^{s}}{a}\partial_{y}(N(e^{-s}v_{y}))+h\right)\,dy$		(3.141)
	$\displaystyle\leq\eta_{1}^{\prime}\int_{\mathbb{R}}y^{2}f^{2}\,dy+E_{11}^{(1)}(s)+C_{2}^{\prime}(\eta_{1}^{\prime})e^{-\delta s}\int_{\mathbb{R}}y^{2}g^{2}\,dy$		(3.142)
	$\displaystyle\quad+C\int_{\mathbb{R}}f_{y}^{2}\,dy+C\left(\frac{e^{-s}}{a}\right)^{2}\int_{\mathbb{R}}f_{yy}^{2}\,dy+C\int_{\mathbb{R}}f^{2}\,dy$		(3.143)
	$\displaystyle\quad+C\left(\\|yh(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|y\partial_{y}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right)$		(3.144)
	$\displaystyle\leq\eta_{1}^{\prime}\int_{\mathbb{R}}y^{2}f^{2}\,dy+E_{11}^{(1)}(s)+C_{2}^{\prime}(\eta_{1}^{\prime})e^{-\delta s}\int_{\mathbb{R}}y^{2}g^{2}\,dy+C\mathbb{E}_{1}^{(0)}(s)$		(3.145)
	$\displaystyle\quad+C\left(\\|yh(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|y\partial_{y}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right)$		(3.146)

for arbitrary small $\eta_{1}^{\prime}>0$ determined later and some constant $C_{2}^{\prime}(\eta_{1}^{\prime})>0$ . Therefore, taking $c_{1}^{(1)}$ and $s_{1}^{(1)}$ so that $c_{1}^{(1)}-(1+C_{1}^{\prime})\geq\frac{1}{2}$ and $c_{1}^{(1)}C_{2}^{\prime}(\eta_{1}^{\prime})e^{-\delta s}\leq\frac{1}{4}$ holds for any $s\geq s_{1}^{(1)}$ , we conclude

	$\displaystyle\frac{d}{ds}\mathbb{E}_{1}^{(1)}+\frac{1}{2}\mathbb{E}_{1}^{(1)}+\frac{1}{4}\int_{\mathbb{R}}y^{2}g^{2}\,dy$		(3.147)
	$\displaystyle\leq\eta_{1}^{\prime}c_{1}^{(1)}\int_{\mathbb{R}}y^{2}f^{2}\,dy-4\frac{e^{-s}}{a}\int_{\mathbb{R}}yf_{yy}g_{y}\,dy+C\int_{\mathbb{R}}f_{y}^{2}\,dy+Ce^{-\delta s}\int_{\mathbb{R}}g^{2}\,dy$		(3.148)
	$\displaystyle\quad+C\mathbb{E}_{1}^{(0)}(s)+C\left(\\|yh(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|y\partial_{y}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right).$		(3.149)

Taking $\eta_{1}^{\prime}$ so that the first term of the right-hand side is bounded by $\eta^{\prime}\mathbb{E}_{1}^{(1)}(s)$ and using $\int_{\mathbb{R}}f_{y}^{2}\,dy\leq C\mathbb{E}_{1}^{(0)}(s)$ , we complete the proof of the second assertion. The first assertion is proved in the same way as before and we omit the detail. ∎

Next, we define

\displaystyle\mathbb{E}_{2}(s):=E_{21}(s)+c_{2}E_{22}(s),

(3.150)

where $c_{2}$ is a sufficiently large constant determined later.

Lemma 3.11.

There exist positive constants $c_{2}$ and $s_{2}$ such that for any $s\geq s_{2}$ , we have

\displaystyle\mathbb{E}_{2}(s)\geq C\left(\int_{\mathbb{R}}f_{yy}^{2}\,dy+\frac{e^{-s}}{2a}\int_{\mathbb{R}}f_{yyy}^{2}\,dy+\frac{r^{2}e^{-s}}{2a}\int_{\mathbb{R}}g_{y}^{2}\,dy+\int_{\mathbb{R}}f_{y}^{2}\,dy\right)

(3.151)

and

	$\displaystyle\frac{d}{ds}\mathbb{E}_{2}(s)+\frac{1}{2}\mathbb{E}_{2}(s)+\frac{1}{4}\int_{\mathbb{R}}g_{y}^{2}\,dy$		(3.152)
	$\displaystyle\leq C\mathbb{E}_{1}^{(0)}(s)+C\left(\\|\partial_{y}h(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}^{2}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right).$		(3.153)

Proof.

By Lemmas 3.5 and 3.7 and the Schwarz inequality, we have

$\displaystyle\frac{d}{ds}E_{21}(s)+\int_{\mathbb{R}}g_{y}^{2}\,dy$	$\displaystyle\leq\frac{5}{2}E_{21}(s)+C_{1}\frac{e^{-s}}{a}\int_{\mathbb{R}}f_{yyy}^{2}\,dy+C_{1}\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}g_{y}^{2}\,dy$	(3.154)
	$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}}g_{y}^{2}\,dy+C\left(\\|h_{y}(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}^{2}N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right)$	(3.155)
	$\displaystyle\leq\left(\frac{5}{2}+2C_{1}\right)E_{21}(s)+\frac{1}{2}\int_{\mathbb{R}}g_{y}^{2}\,dy$	(3.156)
	$\displaystyle\quad+C\left(\\|h_{y}(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}^{2}N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right).$	(3.157)

with some $C_{1}>0$ and

	$\displaystyle\frac{d}{ds}E_{22}(s)+\frac{1}{2}E_{22}(s)+2E_{21}(s)$		(3.158)
	$\displaystyle\leq 2E_{22}(s)+C\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}g_{y}^{2}\,dy+C\frac{r^{2}e^{-s}}{a}\int_{\mathbb{R}}f_{y}^{2}\,dy$		(3.159)
	$\displaystyle\quad+C\int_{\mathbb{R}}f_{y}^{2}\,dy+C\left(\\|h_{y}(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}^{2}N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right)$		(3.160)
	$\displaystyle\leq C_{2}e^{-\delta s}\int_{\mathbb{R}}g_{y}^{2}\,dy+C\int_{\mathbb{R}}f_{y}^{2}\,dy$		(3.161)
	$\displaystyle\quad+C\left(\\|h_{y}(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}^{2}N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right)$		(3.162)

with some $C_{2}>0$ . We take $c_{2}$ sufficiently large so that $2c_{2}-\frac{5}{2}-2C_{1}\geq\frac{1}{2}$ . Then, letting $s_{2}$ sufficiently large so that $c_{2}C_{2}e^{-\delta s}\leq\frac{1}{4}$ holds for any $s\geq s_{2}$ , we conclude

	$\displaystyle\frac{d}{ds}\mathbb{E}_{2}(s)+\frac{1}{2}\mathbb{E}_{2}(s)+\frac{1}{4}\int_{\mathbb{R}}g_{y}^{2}\,dy$		(3.163)
	$\displaystyle\leq C\int_{\mathbb{R}}f_{y}^{2}\,dy+C\left(\\|h_{y}(s)\\|_{L^{2}}^{2}+\frac{e^{2s}}{a^{2}}\\|\partial_{y}^{2}N(e^{-s}v_{y})\\|_{L^{2}}^{2}\right)$		(3.164)

for any $s\geq s_{2}$ . This and $\int_{\mathbb{R}}f_{y}^{2}\,dy\leq C\mathbb{E}_{1}^{(0)}(s)$ complete the proof of the second assertion. The first assertion is proved in the same way as before and we omit the detail. ∎

Finally, let us combine the estimates in Lemmas 3.8–3.11. Fix

\displaystyle\lambda\in\left(0,\min\left\{\frac{1}{2},\frac{2(\beta+1)}{\alpha-\beta+1},\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right\}\right)

(3.165)

and let $s_{\ast}^{\prime}=\max\{s_{0},s_{1}^{(0)},s_{1}^{(1)},s_{2}\}$ . We first note that the Schwarz inequality and Lemma 3.7 imply

\displaystyle-4\frac{e^{-s}}{a}\int_{\mathbb{R}}yf_{yy}g_{y}\,dy

\displaystyle\leq\eta^{\prime\prime}\mathbb{E}_{1}^{(1)}(s)+C(\eta^{\prime\prime})e^{-\delta s}\int_{\mathbb{R}}g_{y}^{2}\,dy

(3.166)

for any $\eta^{\prime\prime}>0$ . We take $\eta,\eta_{1}^{\prime}$ in Lemmas 3.8 and 3.10 and $\eta^{\prime\prime}$ above so that $\frac{1}{2}-\eta\leq\lambda$ and $\eta^{\prime}+\eta^{\prime\prime}\leq\frac{1}{2}-\lambda$ . Then, we take $\tilde{c}_{0}\gg\tilde{c}_{1}^{(0)}\gg\tilde{c}_{1}^{(1)}\gg 1$ and define

$\displaystyle\mathcal{E}(s)$	$\displaystyle=\tilde{c}_{0}\mathbb{E}_{0}(s)+\tilde{c}_{1}^{(0)}\mathbb{E}_{1}^{(0)}(s)+\tilde{c}_{1}^{(1)}\mathbb{E}_{1}^{(1)}(s)+\mathbb{E}_{2}(s)+E_{m1}(s),$	(3.167)
$\displaystyle\mathcal{G}(s)$	$\displaystyle=\int_{\mathbb{R}}\left(\tilde{c}_{0}G^{2}+\tilde{c}_{1}^{(0)}g^{2}+\tilde{c}_{1}^{(1)}y^{2}g^{2}+g_{y}^{2}\right)\,dy,$	(3.168)
$\displaystyle\widetilde{\mathcal{E}}(s)$	$\displaystyle=\mathcal{E}(s)+E_{m2}(s).$	(3.169)

Then, adding the estimates in Lemmas 3.6 and 3.8–3.11, we conclude that

	$\displaystyle\frac{d}{ds}\mathcal{E}(s)+\lambda\mathcal{E}(s)+\frac{1}{4}\mathcal{G}(s)+m_{s}(s)^{2}$		(3.170)
	$\displaystyle\leq Ce^{-\delta s}\int_{\mathbb{R}}g^{2}\,dy+Ce^{-\delta s}\int_{\mathbb{R}}g_{y}^{2}\,dy+\left(\frac{3r^{2}e^{-s}}{4a}-\frac{ra^{\prime}}{2a^{2}}\right)m_{s}(s)^{2}$		(3.171)
	$\displaystyle\quad+C\left(\\|H(s)\\|_{L^{2}}^{2}+\\|h(s)\\|_{H^{0,1}}^{2}+\\|h_{y}(s)\\|_{L^{2}}^{2}\right)$		(3.172)
	$\displaystyle\quad+C\left(\frac{e^{s}}{a}\right)^{2}\left(\\|N(e^{-s}v_{y})\\|_{L^{2}}^{2}+\left\\|\partial_{y}N\left(e^{-s}v_{y}\right)\right\\|_{H^{0,1}}^{2}+\\|\partial_{y}^{2}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right)$		(3.173)

holds for $s\geq s_{\ast}^{\prime}$ . Moreover, Lemma 3.7 leads to

\displaystyle\left(\frac{3r^{2}e^{-s}}{4a}-\frac{ra^{\prime}}{2a^{2}}\right)\leq Ce^{-\delta s}.

(3.174)

Therefore, we finally reach the following energy estimate.

Proposition 3.12.

There exist constants $s_{\ast}>0$ and $C>0$ such that for any $s\geq s_{\ast}$ , we have

	$\displaystyle\frac{d}{ds}\mathcal{E}(s)+\lambda\mathcal{E}(s)+\frac{1}{8}\left(\mathcal{G}(s)+m_{s}^{2}\right)$		(3.175)
	$\displaystyle\leq C\left(\\|H(s)\\|_{L^{2}}^{2}+\\|h(s)\\|_{H^{0,1}}^{2}+\\|h_{y}(s)\\|_{L^{2}}^{2}\right)$		(3.176)
	$\displaystyle\quad+C\left(\frac{e^{s}}{a}\right)^{2}\left(\\|N(e^{-s}v_{y})\\|_{L^{2}}^{2}+\left\\|\partial_{y}N\left(e^{-s}v_{y}\right)\right\\|_{H^{0,1}}^{2}+\\|\partial_{y}^{2}(N(e^{-s}v_{y}))\\|_{L^{2}}^{2}\right).$		(3.177)

4. Estimates of remainder terms and the proof of a priori estimate

In this section, we give estimates of the right-hand side of Proposition 3.12, and complete the a priori estimate, which ensures the existence of the global solution.

4.1. Estimates of remainder terms

First, by the Hardy-type inequality in Lemma 3.2, we have

\displaystyle\|H(s)\|_{L^{2}}^{2}\leq 4\|yh(s)\|_{L^{2}}^{2},\quad\|N(e^{-s}v_{y})\|_{L^{2}}^{2}\leq 4\|y\partial_{y}N(e^{-s}v_{y})\|_{L^{2}}^{2}.

(4.1)

Hence, it suffices to estimate

\displaystyle\|h(s)\|_{H^{0,1}}^{2},\quad\|h_{y}(s)\|_{L^{2}}^{2},\quad\left(\frac{e^{s}}{a}\right)^{2}\left\|\partial_{y}N\left(e^{-s}v_{y}\right)\right\|_{H^{0,1}}^{2},\quad\left(\frac{e^{s}}{a}\right)^{2}\|\partial_{y}^{2}(N(e^{-s}v_{y}))\|_{L^{2}}^{2}.

(4.2)

First, from the definition of $h$ (see (2.13)) and Lemma 3.7, we easily obtain

\displaystyle\|h(s)\|_{H^{0,1}}^{2}+\|h_{y}(s)\|_{L^{2}}^{2}

\displaystyle\leq Ce^{-2\delta s}\left(m(s)^{2}+m_{s}(s)^{2}\right)\leq e^{-2\delta s}\widetilde{\mathcal{E}}(s).

(4.3)

Next, we estimate the nonlinear term. By Assumption (N), we see that

	$\displaystyle\partial_{y}N(e^{-s}v_{y})$	$\displaystyle=2\mu e^{-2s}v_{y}v_{yy}+\tilde{N}^{\prime}(e^{-s}v_{y})e^{-s}v_{yy},$		(4.4)
	$\displaystyle\partial_{y}^{2}N(e^{-s}v_{y})$	$\displaystyle=2\mu e^{-2s}(v_{yy}^{2}+v_{y}v_{yyy})+\tilde{N}^{\prime\prime}(e^{-s}v_{y})e^{-2s}v_{yy}^{2}+\tilde{N}^{\prime}(e^{-s}v_{y})e^{-s}v_{yyy}.$		(4.5)

Therefore, by $|\tilde{N}^{\prime}(z)|\leq C|z|^{p-1}$ , the Sobolev embedding theorem, and

\|f\|_{H^{2,1}}^{2}\leq C\left(\frac{e^{-s}}{a}\right)^{-1}\big{(}\mathbb{E}_{1}^{(0)}(s)+\mathbb{E}_{1}^{(1)}(s)\big{)}\leq Ce^{\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\widetilde{\mathcal{E}}(s),

we have

		$\displaystyle\left(\frac{e^{s}}{a}\right)^{2}\left\\|\partial_{y}N\left(e^{-s}v_{y}\right)\right\\|_{H^{0,1}}^{2}$		(4.6)
		$\displaystyle\leq C\left(\frac{e^{s}}{a}\right)^{2}e^{-4s}\\|v_{y}v_{yy}\\|_{H^{0,1}}^{2}+C\left(\frac{e^{s}}{a}\right)^{2}e^{-2ps}\\|\|v_{y}\|^{p-1}v_{yy}\\|_{H^{0,1}}^{2}$		(4.7)
		$\displaystyle\leq Ce^{-2(1+\frac{\alpha}{\alpha-\beta+1})s}\\|v_{y}\\|_{L^{\infty}}^{2}\\|v_{yy}\\|_{H^{0,1}}^{2}+Ce^{-2(p-2)s}e^{-2(1+\frac{\alpha}{\alpha-\beta+1})s}\\|v_{y}\\|_{L^{\infty}}^{2(p-1)}\\|v_{yy}\\|_{H^{0,1}}^{2}$		(4.8)
		$\displaystyle\leq Ce^{-\frac{2(2\alpha-\beta+1)}{\alpha-\beta+1}s}\\|v_{y}\\|_{H^{1}}^{2}\\|v_{yy}\\|_{H^{0,1}}^{2}+Ce^{-2(p-2)s}e^{-\frac{2(2\alpha-\beta+1)}{\alpha-\beta+1}s}\\|v_{y}\\|_{H^{1}}^{2(p-1)}\\|v_{yy}\\|_{H^{0,1}}^{2}$		(4.9)
		$\displaystyle\leq C\left(e^{-\frac{2(2\alpha-\beta+1)}{\alpha-\beta+1}s}(\\|f\\|_{H^{2,0}}^{2}+m(s)^{2})+e^{-2(p-2)s}e^{-\frac{2(2\alpha-\beta+1)}{\alpha-\beta+1}s}(\\|f\\|_{H^{2,0}}^{2}+m(s)^{2})^{p-1}\right)$		(4.10)
		$\displaystyle\quad\times(\\|f\\|_{H^{2,1}}^{2}+m(s)^{2})$		(4.11)
		$\displaystyle\leq Ce^{-\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\widetilde{\mathcal{E}}(s)^{2}+e^{-\left[2(p-2)+\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right]s}\widetilde{\mathcal{E}}(s)^{p}.$		(4.12)

Similarly, by $|\tilde{N}^{\prime\prime}(z)|\leq C|z|^{p-2}$ , the Sobolev embedding theorem, and

\|f\|_{H^{3,0}}^{2}\leq C\left(\frac{e^{-s}}{a}\right)^{-1}\big{(}\mathbb{E}_{1}^{(0)}(s)+{E}_{21}(s)\big{)}\leq Ce^{\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\widetilde{\mathcal{E}}(s),

we obtain

		$\displaystyle\left(\frac{e^{s}}{a}\right)^{2}\left\\|\partial_{y}^{2}N\left(e^{-s}v_{y}\right)\right\\|_{L^{2}}^{2}$		(4.13)
		$\displaystyle\leq C\left(\frac{e^{s}}{a}\right)^{2}e^{-4s}\left(\\|v_{yy}^{2}\\|_{L^{2}}^{2}+\\|v_{y}v_{yyy}\\|_{L^{2}}^{2}\right)$		(4.14)
		$\displaystyle\quad+C\left(\frac{e^{s}}{a}\right)^{2}e^{-2ps}\left(\\|\|v_{y}\|^{p-2}v_{yy}^{2}\\|_{L^{2}}^{2}+\\|\|v_{y}\|^{p-1}v_{yyy}\\|_{L^{2}}^{2}\right)$		(4.15)
		$\displaystyle\leq Ce^{-2(1+\frac{\alpha}{\alpha-\beta+1})s}(\\|v_{yy}\\|_{L^{\infty}}^{2}\\|v_{yy}\\|_{L^{2}}^{2}+\\|v_{y}\\|_{L^{\infty}}^{2}\\|v_{yyy}\\|_{L^{2}}^{2})$		(4.16)
		$\displaystyle\quad+Ce^{-2(p-2)s}e^{-2(1+\frac{\alpha}{\alpha-\beta+1})s}(\\|v_{y}\\|_{L^{\infty}}^{2(p-2)}\\|v_{yy}\\|_{L^{\infty}}^{2}\\|v_{yy}\\|_{L^{2}}^{2}+\\|v_{y}\\|_{L^{\infty}}^{2(p-1)}\\|v_{yyy}\\|_{L^{2}}^{2})$		(4.17)
		$\displaystyle\leq Ce^{-\frac{2(2\alpha-\beta+1)}{\alpha-\beta+1})s}(\\|v_{yy}\\|_{H^{1,0}}^{2}\\|v_{yy}\\|_{L^{2}}^{2}+\\|v_{y}\\|_{H^{1,0}}^{2}\\|v_{yyy}\\|_{L^{2}}^{2})$		(4.18)
		$\displaystyle\quad+Ce^{-2(p-2)s}e^{-\frac{2(2\alpha-\beta+1)}{\alpha-\beta+1})s}(\\|v_{y}\\|_{H^{1,0}}^{2(p-2)}\\|v_{yy}\\|_{L^{\infty}}^{2}\\|v_{yy}\\|_{L^{2}}^{2}+\\|v_{y}\\|_{H^{1,0}}^{2(p-1)}\\|v_{yyy}\\|_{L^{2}}^{2})$		(4.19)
		$\displaystyle\leq C\left(e^{-\frac{2(2\alpha-\beta+1)}{\alpha-\beta+1})s}(\\|f\\|_{H^{2,0}}^{2}+m(s)^{2})+e^{-2(p-2)s}e^{-\frac{2(2\alpha-\beta+1)}{\alpha-\beta+1})s}(\\|f\\|_{H^{2,0}}^{2}+m(s)^{2})^{p-1}\right)$		(4.20)
		$\displaystyle\quad\times(\\|f\\|_{H^{3,0}}^{2}+m(s)^{2})$		(4.21)
		$\displaystyle\leq Ce^{-\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\widetilde{\mathcal{E}}(s)^{2}+e^{-\left[2(p-2)+\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right]s}\widetilde{\mathcal{E}}(s)^{p}.$		(4.22)

4.2. Proof of a priori estimate

Combining the energy estimates obtained in Proposition 3.12 with the estimates of remainder terms given in the previous subsection, we deduce

		$\displaystyle\frac{d}{ds}\mathcal{E}(s)+\lambda\mathcal{E}(s)+\frac{1}{8}\left(\mathcal{G}(s)+m_{s}(s)^{2}\right)$		(4.23)
		$\displaystyle\leq Ce^{-2\delta s}\widetilde{\mathcal{E}}(s)+Ce^{-\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\widetilde{\mathcal{E}}(s)^{2}+Ce^{-\left[2(p-2)+\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right]s}\widetilde{\mathcal{E}}(s)^{p}$		(4.24)

From Lemmas 3.6 and 3.7, we see that

	$\displaystyle\frac{d}{ds}E_{m2}(s)$	$\displaystyle=2E_{m1}(s)+\left(\frac{r^{\prime}}{a}-\frac{ra^{\prime}}{a^{2}}\right)m(s)m_{s}(s)$		(4.25)
		$\displaystyle\leq Ce^{-\delta s}m_{s}(s)^{2}+\frac{1}{16}m_{s}(s)^{2}+Ce^{-2\delta s}m(s)^{2}.$		(4.26)

Therefore, there exists constants $s_{m}\geq s_{\ast}$ and $c>0$ such that for any $s\geq s_{m}$ , we have

		$\displaystyle\frac{d}{ds}\widetilde{\mathcal{E}}(s)+\lambda\widetilde{\mathcal{E}}(s)+c(\mathcal{G}(s)+m_{s}(s)^{2})$		(4.27)
		$\displaystyle\leq C_{3}e^{-2\delta s}\widetilde{\mathcal{E}}(s)+C_{3}\left(e^{-\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\widetilde{\mathcal{E}}(s)^{2}+e^{-\left[2(p-2)+\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right]s}\widetilde{\mathcal{E}}(s)^{p}\right)$		(4.28)

with some $C_{3}>0$ . Define

\displaystyle\Lambda(s)=\exp\left(-C_{3}\int_{s_{m}}^{s}e^{-2\delta\sigma}\,d\sigma\right).

(4.29)

Note that

\displaystyle\Lambda(s)=\exp\left(\frac{C_{3}}{2\delta}\left(e^{-2\delta s}-e^{-2\delta s_{m}}\right)\right)\sim 1\quad\text{and}\quad\Lambda(s_{m})=1.

(4.30)

Multiplying (4.28) by $\Lambda(s)$ , we deduce

	$\displaystyle\frac{d}{ds}\left[\Lambda(s)\widetilde{\mathcal{E}}(s)\right]+\lambda\Lambda(s)\mathcal{E}(s)+c\Lambda(s)\left(\mathcal{G}(s)+m_{s}(s)^{2}\right)$		(4.31)
	$\displaystyle\leq C_{3}\Lambda(s)\left(e^{-\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\widetilde{\mathcal{E}}(s)^{2}+e^{-\left[2(p-2)+\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right]s}\widetilde{\mathcal{E}}(s)^{p}\right).$		(4.32)

Integrating the above over $[s_{m},s]$ , we have

\displaystyle\Lambda(s)\widetilde{\mathcal{E}}(s)

\displaystyle\leq\widetilde{\mathcal{E}}(s_{m})+C_{3}\int_{s_{m}}^{s}\Lambda(\sigma)\left(e^{-\frac{2\alpha-\beta+1}{\alpha-\beta+1}s}\widetilde{\mathcal{E}}(\sigma)^{2}+e^{-\left[2(p-2)+\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right]s}\widetilde{\mathcal{E}}(\sigma)^{p}\right)\,d\sigma

(4.33)

Finally, we put

\displaystyle\widetilde{\mathcal{E}}_{\max}(s)=\max_{\sigma\in[s_{m},s]}\widetilde{\mathcal{E}}(\sigma)

(4.34)

for $s\geq s_{m}$ . Then, the above estimate implies

\displaystyle\widetilde{\mathcal{E}}_{\max}(s)

\displaystyle\leq C_{0}\widetilde{\mathcal{E}}(s_{m})+C_{0}^{\prime}\left(\widetilde{\mathcal{E}}_{\max}(s)^{2}+\widetilde{\mathcal{E}}_{\max}(s)^{p}\right)

(4.35)

with some constants $C_{0},C_{0}^{\prime}>0$ , where we have used $\delta>0$ and $p>\frac{-\beta+1}{\alpha-\beta+1}$ (see Remark 1.1). Thus, we conclude the a priori estimate

\displaystyle\widetilde{\mathcal{E}}_{\max}(s)\leq 2C_{0}\widetilde{\mathcal{E}}(s_{m})

(4.36)

for all $s\geq s_{m}$ , provided that $\widetilde{\mathcal{E}}(s_{m})$ is sufficiently small. From the local existence result (Proposition B.2), we see that, for sufficiently small initial data, the local solution uniquely exists over $[0,s_{m}]$ , and it satisfies $\widetilde{\mathcal{E}}(s_{m})\leq C(\|u_{0}\|_{H^{2,1}\cap H^{3,0}}+\|u_{1}\|_{H^{0,1}\cap H^{1,0}})$ (for the detail, see the proof of Proposition B.2 (vi) ). Thus, $\widetilde{\mathcal{E}}(s_{m})$ can be controlled by the norm of initial data. This and Proposition B.2 (iii) (blow-up alternative) indicate the existence of the global solution if the initial data $(u_{0},u_{1})$ is sufficiently small.

It remains to prove the asymptotic estimate. To this end, we go back to the estimate (4.24). By virtue of the a priori estimate (4.36), we have

	$\displaystyle\frac{d}{ds}\mathcal{E}(s)+\lambda\mathcal{E}(s)+\frac{1}{8}\left(\mathcal{G}(s)+m_{s}(s)^{2}\right)$	$\displaystyle\leq Ce^{-\min\{2\delta,\frac{2\alpha-\beta+1}{\alpha-\beta+1},2(p-2)+\frac{2\alpha-\beta+1}{\alpha-\beta+1}\}s}\widetilde{\mathcal{E}}(s_{m})$		(4.37)
		$\displaystyle=Ce^{-\min\{\frac{2(\beta+1)}{\alpha-\beta+1},\frac{2\alpha-\beta+1}{\alpha-\beta+1}\}s}\widetilde{\mathcal{E}}(s_{m}),$		(4.38)

where we have also used $\widetilde{\mathcal{E}}(s_{m})$ , which can be assumed without loss of generality. Now, recall

\displaystyle\lambda\in\left(0,\min\left\{\frac{1}{2},\frac{2(\beta+1)}{\alpha-\beta+1},\frac{2\alpha-\beta+1}{\alpha-\beta+1}\right\}\right),

(4.39)

and multiply the above estimate by $e^{\lambda s}$ . Then, we obtain

\displaystyle\frac{d}{ds}\left[e^{\lambda s}\mathcal{E}(s)\right]+\frac{1}{8}e^{\lambda s}\left(\mathcal{G}(s)+m_{s}(s)^{2}\right)\leq Ce^{\lambda-\min\{\frac{2(\beta+1)}{\alpha-\beta+1},\frac{2\alpha-\beta+1}{\alpha-\beta+1}\}s}\widetilde{\mathcal{E}}(s_{m}).

(4.40)

Integrating this over $[s_{m},s]$ implies

\displaystyle e^{\lambda s}\mathcal{E}(s)+\frac{1}{8}\int_{s_{m}}^{s}e^{\lambda\sigma}\left(\mathcal{G}(\sigma)+m_{s}(\sigma)^{2}\right)\,d\sigma\leq C\widetilde{\mathcal{E}}(s_{m}).

(4.41)

Therefore, we have

\displaystyle\mathcal{E}(s)\leq Ce^{-\lambda s}\widetilde{\mathcal{E}}(s_{m})

(4.42)

for all $s\geq s_{m}$ . Moreover, we deduce

\displaystyle\int_{s_{m}}^{s}e^{\lambda\sigma}m_{s}(\sigma)^{2}\,d\sigma\leq C\widetilde{\mathcal{E}}(s_{m}).

(4.43)

This shows, for any $s\geq s^{\prime}\geq s_{m}$ ,

$\displaystyle\|m(s)-m(s^{\prime})\|$	$\displaystyle=\left\|\int_{s^{\prime}}^{s}m_{s}(\sigma)\,d\sigma\right\|$	(4.44)
	$\displaystyle\leq\left(\int_{s^{\prime}}^{s}e^{-\lambda\sigma}\,d\sigma\right)^{1/2}\left(\int_{s^{\prime}}^{s}e^{\lambda\sigma}m_{s}(\sigma)^{2}\,d\sigma\right)^{1/2}$	(4.45)
	$\displaystyle\leq\left(\frac{1}{\lambda}(e^{-\lambda s^{\prime}}-e^{-\lambda s})\right)^{1/2}C\widetilde{\mathcal{E}}_{m}(s_{m})^{1/2}$	(4.46)
	$\displaystyle\to 0\quad(s^{\prime},s\to\infty).$	(4.47)

This means that the limit $m^{\ast}=\lim_{s\to\infty}m(s)$ exists and satisfies

\displaystyle|m^{\ast}-m(s)|^{2}\leq C\widetilde{\mathcal{E}}(s_{m})e^{-\lambda s}

(4.48)

for all $s\geq s_{m}$ . Consequently, by the above estimate and (4.42), we have

$\displaystyle\\|v(s)-m^{\ast}\varphi\\|_{L^{2}}^{2}$	$\displaystyle=\\|m(s)\varphi+f(s)-m^{\ast}\varphi\\|_{L^{2}}^{2}$	(4.49)
	$\displaystyle\leq C\left(\|m^{\ast}-m(s)\|^{2}\\|\varphi\\|_{L^{2}}^{2}+\\|f(s)\\|_{L^{2}}^{2}\right)$	(4.50)
	$\displaystyle\leq Ce^{-\lambda s}\widetilde{\mathcal{E}}(s_{m})$	(4.51)
	$\displaystyle\leq Ce^{-\lambda s}\left(\\|u_{0}\\|_{H^{2,1}\cap H^{3,0}}^{2}+\\|u_{1}\\|_{H^{0,1}\cap H^{1,0}}\right)^{2}$	(4.52)

for $s\geq s_{m}$ , which implies

\displaystyle\|u(t)-m^{\ast}G(R(t))\|_{L^{2}}^{2}

\displaystyle\leq C(R(t)+1)^{-\frac{1}{2}-\lambda}\left(\|u_{0}\|_{H^{2,1}\cap H^{3,0}}^{2}+\|u_{1}\|_{H^{0,1}\cap H^{1,0}}\right)^{2}

(4.53)

for $t\geq t_{m}:=R^{-1}(e^{s}-1)$ . This completes the proof of the asymptotic estimate.

Appendix A A general lemma for the energy identity

In this appendix, we give a proof of Lemma 3.1. Actually, we give a slightly more general version of it and prove the following lemma. If we take $k=\frac{1}{2}$ and $c_{3}(s)\equiv 1$ , then we have Lemma 3.1.

Lemma A.1.

Let $k,l,m\in\mathbb{R}$ , $n\in\mathbb{N}\cup\{0\}$ , and let $c_{j}=c_{j}(s)\ (j=1,2,3,4)$ be smooth functions defined on $[0,\infty)$ . We consider a system for two functions $f=f(s,y)$ and $g=g(s,y)$ given by

\displaystyle\left\{\begin{aligned} &f_{s}-kyf_{y}-lf=g,\\ &c_{1}(s)\left(g_{s}-kyg_{y}-mg\right)+c_{2}(s)g+g=c_{3}(s)f_{yy}-c_{4}(s)f_{yyyy}+h\end{aligned}\right.\quad(s,y)\in(0,\infty)\times\mathbb{R},

(A.1)

where $h=h(s,y)$ is a given smooth function belonging to $C([0,\infty);H^{0,n}(\mathbb{R}))$ . We define the energies

	$\displaystyle E_{1}(s)$	$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}y^{2n}\left(c_{3}(s)f_{y}^{2}+c_{4}(s)f_{yy}^{2}+c_{1}(s)g^{2}\right)\,dy,$		(A.2)
	$\displaystyle E_{2}(s)$	$\displaystyle=\int_{\mathbb{R}}y^{2n}\left(\frac{1}{2}f^{2}+c_{1}(s)fg\right)\,dy.$		(A.3)

Then, we have

$\displaystyle\frac{d}{ds}E_{1}(s)$	$\displaystyle=-\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\left(-\frac{2n-1}{2}k+l\right)c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy+\left(-\frac{2n-3}{2}k+l\right)c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy$	(A.4)
	$\displaystyle\quad+\left(-\frac{2n+1}{2}k+m\right)c_{1}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(A.5)
	$\displaystyle\quad-2nc_{3}(s)\int_{\mathbb{R}}y^{2n-1}f_{y}g\,dy-2n(2n-1)c_{4}(s)\int_{\mathbb{R}}y^{2n-2}f_{yy}g\,dy-4nc_{4}(s)\int_{\mathbb{R}}y^{2n-1}f_{yy}g_{y}\,dy$	(A.6)
	$\displaystyle\quad+\frac{c_{3}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy+\frac{c_{4}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+\frac{c_{1}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\int_{\mathbb{R}}y^{2n}gh\,dy$	(A.7)

and

$\displaystyle\frac{d}{ds}E_{2}(s)$	$\displaystyle=-c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy-c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+\left(-\frac{2n+1}{2}k+l\right)\int_{\mathbb{R}}y^{2n}f^{2}\,dy$	(A.8)
	$\displaystyle\quad+c_{1}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\left(-(2n+1)k+l+m\right)c_{1}(s)\int_{\mathbb{R}}y^{2n}fg\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}fg\,dy$	(A.9)
	$\displaystyle\quad-2nc_{3}(s)\int_{\mathbb{R}}y^{2n-1}ff_{y}\,dy-4nc_{4}(s)\int_{\mathbb{R}}y^{2n-1}f_{y}f_{yy}\,dy-2n(2n-1)c_{4}(s)\int_{\mathbb{R}}y^{2n-2}ff_{yy}\,dy$	(A.10)
	$\displaystyle\quad+c_{1}^{\prime}(s)\int_{\mathbb{R}}y^{2n}fg\,dy+\int_{\mathbb{R}}y^{2n}fh\,dy.$	(A.11)

Proof of Lemma A.1.

We calculate

$\displaystyle\frac{d}{ds}E_{1}(s)$	$\displaystyle=\frac{d}{ds}\left[\frac{1}{2}\int_{\mathbb{R}}y^{2n}\left(c_{3}(s)f_{y}^{2}+c_{4}(s)f_{yy}^{2}+c_{1}(s)g^{2}\right)\,dy\right]$	(A.12)
	$\displaystyle=c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}f_{ys}\,dy+\frac{c_{3}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy$	(A.13)
	$\displaystyle\quad+c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}f_{yys}\,dy+\frac{c_{4}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy$	(A.14)
	$\displaystyle\quad+c_{1}(s)\int_{\mathbb{R}}y^{2n}gg_{s}\,dy+\frac{c_{1}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}g^{2}\,dy.$	(A.15)

Using the equation (A.1), we rewrite the above identity as

$\displaystyle\frac{d}{ds}E_{1}(s)$	$\displaystyle=c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}(kyf_{y}+lf+g)_{y}\,dy+\frac{c_{3}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy$	(A.16)
	$\displaystyle\quad+c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}(kyf_{y}+lf+g)_{yy}\,dy+\frac{c_{4}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy$	(A.17)
	$\displaystyle\quad+c_{1}(s)\int_{\mathbb{R}}y^{2n}g(kyg_{y}+mg)\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy-\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(A.18)
	$\displaystyle\quad+c_{3}(s)\int_{\mathbb{R}}y^{2n}gf_{yy}\,dy-c_{4}(s)\int_{\mathbb{R}}y^{2n}gf_{yyyy}\,dy+\int_{\mathbb{R}}y^{2n}gh\,dy$	(A.19)
	$\displaystyle\quad+\frac{c_{1}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}g^{2}\,dy.$	(A.20)

By noting the relations

$\displaystyle y^{2n}f_{y}(yf_{y})_{y}$	$\displaystyle=\left(\frac{y^{2n+1}}{2}f_{y}^{2}\right)_{y}-\frac{2n-1}{2}y^{2n}f_{y}^{2},$	(A.21)
$\displaystyle y^{2n}f_{yy}(yf_{y})_{yy}$	$\displaystyle=\left(\frac{y^{2n+1}}{2}f_{yy}^{2}\right)_{y}-\frac{2n-3}{2}y^{2n}f_{yy}^{2},$	(A.22)
$\displaystyle y^{2n}g(yg_{y})$	$\displaystyle=\left(\frac{y^{2n+1}}{2}g^{2}\right)_{y}-\frac{2n+1}{2}y^{2n}g^{2},$	(A.23)
$\displaystyle y^{2n}gf_{yy}$	$\displaystyle=\left(y^{2n}gf_{y}\right)_{y}-y^{2n}f_{y}g_{y}-2ny^{2n-1}f_{y}g,$	(A.24)
$\displaystyle y^{2n}gf_{yyyy}$	$\displaystyle=\left(y^{2n}gf_{yyy}\right)_{y}-\left((y^{2n}g)_{y}f_{yy}\right)_{y}$	(A.25)
	$\displaystyle\quad+\left(2n(2n-1)y^{2n-2}g+4ny^{2n-1}g_{y}+y^{2n}g_{yy}\right)f_{yy},$	(A.26)

we have

$\displaystyle\frac{d}{ds}E_{1}(s)$	$\displaystyle=\left(-\frac{2n-1}{2}k+l\right)c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy+c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}g_{y}\,dy+\frac{c_{3}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy$	(A.27)
	$\displaystyle\quad+\left(-\frac{2n-3}{2}k+l\right)c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}g_{yy}\,dy+\frac{c_{4}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy$	(A.28)
	$\displaystyle\quad+\left(-\frac{2n+1}{2}k+m\right)c_{1}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy-\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(A.29)
	$\displaystyle\quad-c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}g_{y}\,dy-2nc_{3}(s)\int_{\mathbb{R}}y^{2n-1}f_{y}g\,dy$	(A.30)
	$\displaystyle\quad-2n(2n-1)c_{4}(s)\int_{\mathbb{R}}y^{2n-2}f_{yy}g\,dy-4nc_{4}(s)\int_{\mathbb{R}}y^{2n-1}f_{yy}g_{y}\,dy-c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}g_{yy}\,dy$	(A.31)
	$\displaystyle\quad+\int_{\mathbb{R}}y^{2n}gh\,dy+\frac{c_{1}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}g^{2}\,dy.$	(A.32)

Thus, we conclude

$\displaystyle\frac{d}{ds}E_{1}(s)$	$\displaystyle=-\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\left(-\frac{2n-1}{2}k+l\right)c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy+\left(-\frac{2n-3}{2}k+l\right)c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy$	(A.33)
	$\displaystyle\quad+\left(-\frac{2n+1}{2}k+m\right)c_{1}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(A.34)
	$\displaystyle\quad-2nc_{3}(s)\int_{\mathbb{R}}y^{2n-1}f_{y}g\,dy-2n(2n-1)c_{4}(s)\int_{\mathbb{R}}y^{2n-2}f_{yy}g\,dy-4nc_{4}(s)\int_{\mathbb{R}}y^{2n-1}f_{yy}g_{y}\,dy$	(A.35)
	$\displaystyle\quad+\frac{c_{3}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy+\frac{c_{4}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+\frac{c_{1}^{\prime}(s)}{2}\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\int_{\mathbb{R}}y^{2n}gh\,dy.$	(A.36)

This gives the desired identity for $E_{1}(s)$ . Next, we compute

	$\displaystyle\frac{d}{ds}E_{2}(s)$	$\displaystyle=\frac{d}{ds}\left[\int_{\mathbb{R}}y^{2n}\left(\frac{1}{2}f^{2}+c_{1}(s)fg\right)\,dy\right]$		(A.37)
		$\displaystyle=\int_{\mathbb{R}}y^{2n}ff_{s}\,dy+c_{1}(s)\int_{\mathbb{R}}y^{2n}f_{s}g\,dy+c_{1}(s)\int_{\mathbb{R}}y^{2n}fg_{s}\,dy+c_{1}^{\prime}(s)\int_{\mathbb{R}}y^{2n}fg\,dy.$		(A.38)

Using the equation (A.1), we rewrite the above identity as

$\displaystyle\frac{d}{ds}E_{2}(s)$	$\displaystyle=\int_{\mathbb{R}}y^{2n}f(kyf_{y}+lf+g)\,dy+c_{1}(s)\int_{\mathbb{R}}y^{2n}(kyf_{y}+lf+g)g\,dy$	(A.39)
	$\displaystyle\quad+c_{1}(s)\int_{\mathbb{R}}y^{2n}f(kyg_{y}+mg)\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}fg\,dy-\int_{\mathbb{R}}y^{2n}fg\,dy$	(A.40)
	$\displaystyle\quad+c_{3}(s)\int_{\mathbb{R}}y^{2n}ff_{yy}\,dy-c_{4}(s)\int_{\mathbb{R}}y^{2n}ff_{yyyy}\,dy$	(A.41)
	$\displaystyle\quad+\int_{\mathbb{R}}y^{2n}fh\,dy+c_{1}^{\prime}(s)\int_{\mathbb{R}}y^{2n}fg\,dy.$	(A.42)

By noting the relations

$\displaystyle y^{2n}f(yf_{y})$	$\displaystyle=\left(\frac{y^{2n+1}}{2}f^{2}\right)_{y}-\frac{2n+1}{2}y^{2n}f^{2},$	(A.43)
$\displaystyle y^{2n}f(yg_{y})$	$\displaystyle=\left(y^{2n+1}fg\right)_{y}-y^{2n+1}f_{y}g-(2n+1)y^{2n}fg,$	(A.44)
$\displaystyle y^{2n}ff_{yy}$	$\displaystyle=\left(y^{2n}ff_{y}\right)_{y}-y^{2n}f_{y}^{2}-2ny^{2n-1}ff_{y},$	(A.45)
$\displaystyle y^{2n}ff_{yyyy}$	$\displaystyle=\left(y^{2n}ff_{yyy}\right)_{y}-\left((y^{2n}f)_{y}f_{yy}\right)_{y}$	(A.46)
	$\displaystyle\quad+(2n(2n-1)y^{2n-2}f+4ny^{2n-1}f_{y}+y^{2n}f_{yy})f_{yy},$	(A.47)

we have

$\displaystyle\frac{d}{ds}E_{2}(s)$	$\displaystyle=\left(-\frac{2n+1}{2}k+l\right)\int_{\mathbb{R}}y^{2n}f^{2}\,dy+\int_{\mathbb{R}}y^{2n}fg\,dy$	(A.48)
	$\displaystyle\quad+kc_{1}(s)\int_{\mathbb{R}}y^{2n+1}f_{y}g\,dy+lc_{1}(s)\int_{\mathbb{R}}y^{2n}fg\,dy+c_{1}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy$	(A.49)
	$\displaystyle\quad-kc_{1}(s)\int_{\mathbb{R}}y^{2n+1}f_{y}g\,dy+\left(-(2n+1)k+m\right)c_{1}(s)\int_{\mathbb{R}}y^{2n}fg\,dy$	(A.50)
	$\displaystyle\quad-c_{2}(s)\int_{\mathbb{R}}y^{2n}fg\,dy-\int_{\mathbb{R}}y^{2n}fg\,dy$	(A.51)
	$\displaystyle\quad-c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy-2nc_{3}(s)\int_{\mathbb{R}}y^{2n-1}ff_{y}\,dy$	(A.52)
	$\displaystyle\quad-2n(2n-1)c_{4}(s)\int_{\mathbb{R}}y^{2n-2}ff_{yy}\,dy-4nc_{4}(s)\int_{\mathbb{R}}y^{2n-1}f_{y}f_{yy}\,dy-c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy$	(A.53)
	$\displaystyle\quad+\int_{\mathbb{R}}y^{2n}fh\,dy+c_{1}^{\prime}(s)\int_{\mathbb{R}}y^{2n}fg\,dy.$	(A.54)

Thus, we conclude

$\displaystyle\frac{d}{ds}E_{2}(s)$	$\displaystyle=-c_{3}(s)\int_{\mathbb{R}}y^{2n}f_{y}^{2}\,dy-c_{4}(s)\int_{\mathbb{R}}y^{2n}f_{yy}^{2}\,dy+\left(-\frac{2n+1}{2}k+l\right)\int_{\mathbb{R}}y^{2n}f^{2}\,dy$	(A.55)
	$\displaystyle\quad+c_{1}(s)\int_{\mathbb{R}}y^{2n}g^{2}\,dy+\left(-(2n+1)k+l+m\right)c_{1}(s)\int_{\mathbb{R}}y^{2n}fg\,dy-c_{2}(s)\int_{\mathbb{R}}y^{2n}fg\,dy$	(A.56)
	$\displaystyle\quad-2nc_{3}(s)\int_{\mathbb{R}}y^{2n-1}ff_{y}\,dy-4nc_{4}(s)\int_{\mathbb{R}}y^{2n-1}f_{y}f_{yy}\,dy-2n(2n-1)c_{4}(s)\int_{\mathbb{R}}y^{2n-2}ff_{yy}\,dy$	(A.57)
	$\displaystyle\quad+c_{1}^{\prime}(s)\int_{\mathbb{R}}y^{2n}fg\,dy+\int_{\mathbb{R}}y^{2n}fh\,dy.$	(A.58)

This completes the proof. ∎

Appendix B Local existence

We discuss the local existence and basic properties of solutions to (1.1). Let $X=H^{3,0}(\mathbb{R})\times H^{1,0}(\mathbb{R})$ and

U:=\begin{pmatrix}u\\ \partial_{t}u\end{pmatrix},\quad U_{0}:=\begin{pmatrix}u_{0}\\ u_{1}\end{pmatrix}.

Let $D(A)=H^{5,0}(\mathbb{R})\times H^{3,0}(\mathbb{R})$ and define

A=\begin{pmatrix}0&1\\ -\partial_{x}^{4}&0\end{pmatrix},\quad\mathcal{T}(t)=\exp(tA).

We also define

K(\sigma;U(s))=\begin{pmatrix}0\\ -b(\sigma)\partial_{t}u(s)+a(\sigma)\partial_{x}^{2}u(s)+\partial_{x}N(\partial_{x}u(s))\end{pmatrix},

namely, $\sigma$ and $s$ denote the variables for the coefficients $a(t),b(t)$ and the unknown $u$ , respectively.

Now, we introduce the definition of the strong solution and the mild solution.

Definition B.1.

Let $I=[0,T]$ with some $T>0$ or $I=[0,\infty)$ . We say that a function $u$ (or $U={}^{t}(u,\partial_{t}u)$ ) is a strong solution to (1.1) on $I$ if

\left\{\begin{aligned} &U\in C(I;D(A))\cap C^{1}(I;X),\\ &\dfrac{d}{dt}U(t)=AU(t)+K(t;U(t))\quad\text{on}\ I,\\ &U(0)=U_{0}.\end{aligned}\right.

Also, we say that a function $u$ (or $U={}^{t}(u,\partial_{t}u)$ ) is a mild solution to (1.1) on $I$ if

\left\{\begin{aligned} &U\in C(I;X),\\ &U(t)=\mathcal{T}(t)U_{0}+\int_{0}^{t}\mathcal{T}(t-s)K(s;U(s))\,ds\quad\text{in}\ C(I;X).\\ \end{aligned}\right.

Proposition B.2.

(i) (Local existence) For any $U_{0}\in X$ , there exists $T>0$ such that there exists a mild solution to (1.1) on $[0,T]$ .

(ii) (Uniqueness) Let $T>0$ . If $U$ and $V$ are mild solutions in $C([0,T];X)$ with the same initial condition $U(0)=V(0)=U_{0}$ , then $U=V$ .

(iii) (Blow-up alternative) Let $T_{\max}=T_{\max}(U_{0})$ be

T_{\max}=\sup\{T\in(0,\infty];\,\exists U\in C([0,T];X):\text{a mild solution to \eqref{eq:ndb}}\}.

If $T_{\max}<\infty$ , then $\lim_{t\to T_{\max-0}}\|U(t)\|_{X}=\infty$ .

(iv) (Continuous dependence on the initial data) Let $U_{0}\in X$ and $\{U_{0}^{(j)}\}_{j=1}^{\infty}$ a sequence in $X$ satisfying $\lim_{j\to\infty}\|U_{0}^{(j)}-U_{0}\|_{X}=0$ . Let $U$ and $U^{(j)}$ be the corresponding mild solutions to the initial data $U_{0}$ and $U_{0}^{(j)}$ , respectively. Then, for any fixed $T\in(0,T_{\max}(U_{0}))$ , we have $T_{\max}(U_{0}^{(j)})>T$ for sufficiently large $j$ and

\lim_{j\to\infty}\sup_{t\in[0,T]}\|U^{(j)}(t)-U(t)\|_{X}=0.

(v) (Regularity) Let $T>0$ . If $U_{0}\in D(A)$ , then the mild solution in (i) on $[0,T]$ becomes a strong solution on $[0,T]$ .

(vi) (Small data almost global existence) For any $T>0$ , there exists $\varepsilon_{0}>0$ such that if $\|U_{0}\|_{X}<\varepsilon_{0}$ , then the corresponding mild solution $U$ can be extended to $[0,T]$ .

(vii) (Boundedness of weighted norm) Let $T>0$ and $Y:=H^{2,1}(\mathbb{R})\times H^{0,1}(\mathbb{R})$ . If $U_{0}\in X\cap Y$ , then the corresponding mild solution $U$ on $[0,T]$ belongs to $C([0,T];X\cap Y)$ .

Proof.

Let $T_{0}>0$ be fixed. Then, $a(t),b(t)$ are positive, and they and their first derivatives are bounded by some constant $C_{T_{0}}>0$ on $[0,T_{0}]$ . Let $T\in(0,T_{0}]$ . Then, for any $U={}^{t}(u,v)\in X$ and $t\in[0,T]$ , we have

$\displaystyle\\|K(t;U)\\|_{X}$	$\displaystyle=\left\\|-b(t)\partial_{t}u+a(t)\partial_{x}^{2}u+\partial_{x}N(\partial_{x}u)\right\\|_{H^{1}}$	(B.1)
	$\displaystyle\leq C_{T_{0}}\left(\\|\partial_{t}u\\|_{H^{1}}+\\|\partial_{x}^{2}u\\|_{H^{1}}\right)$	(B.2)
	$\displaystyle\quad+C\left(\\|\partial_{x}u\\|_{W^{1,\infty}}+\\|\partial_{x}u\\|_{W^{1,\infty}}^{p-1}\right)\\|\partial_{x}^{2}u\\|_{H^{1}}<\infty,$	(B.3)

that is, $K(t;\cdot):X\to X$ . Moreover, for $M>0$ and $U={}^{t}(u,v),W={}^{t}(w,z)\in B_{M}=\{U\in X;\|U\|_{X}\leq M\}$ , we calculate

$\displaystyle\\|K(t;U)-K(t;W)\\|_{X}$	$\displaystyle\leq C_{T_{0}}\left(\\|v-z\\|_{H^{1}}+\\|u-w\\|_{H^{3}}\right)$	(B.4)
	$\displaystyle\quad+C\left(\\|u\\|_{W^{2,\infty}}+\\|w\\|_{W^{2,\infty}}\right)\\|u-w\\|_{H^{3}}$	(B.5)
	$\displaystyle\quad+C\left(\|u\\|_{W^{2,\infty}}+\\|w\\|_{W^{2,\infty}}\right)^{p-1}\\|u-w\\|_{H^{3}}$	(B.6)
	$\displaystyle\leq C_{T_{0},M}\\|U-W\\|_{X}.$	(B.7)

Therefore, $K(t;\cdot)$ is locally Lipschitz continuous in $X$ . Therefore, from the proofs of [3, Lemmas 4.3.2, Proposition 4.3.3], there exist $T>0$ and a unique mild solution $u$ on $I=[0,T]$ . Also, [3, Theorem 4.3.4] shows the property (iii). Moreover, by [3, Proposition 4.3.7], the continuous dependence on the initial data. This proves (i)–(iv).

Next, we prove (iv) along with the argument of [3, Lemma 4.3.9]. Take $U_{0}\in D(A)$ and $T\in(0,T_{\max})$ . Let $h>0$ , $t\in[0,T-h]$ , and $M:=\sup_{s\in[0,T]}\|U(s)\|_{X}$ . Consider

$\displaystyle U(t+h)-U(t)$	$\displaystyle=\mathcal{T}(h)\mathcal{T}(t)U_{0}-\mathcal{T}(t)U_{0}$	(B.8)
	$\displaystyle\quad+\int_{0}^{t}\mathcal{T}(s)\left\{K(t+h-s;U(t+h-s))-K(t-s;U(t-s))\right)\}\,ds$	(B.9)
	$\displaystyle\quad+\int_{0}^{h}\mathcal{T}(t+s)K(h-s;U(h-s))\,ds$	(B.10)
	$\displaystyle=:J_{1}+J_{2}+J_{3}.$	(B.11)

For $J_{1},J_{2}$ , we estimate

	$\displaystyle\\|J_{1}\\|_{X}$	$\displaystyle\leq\\|\mathcal{T}(h)U_{0}-U_{0}\\|_{X}=\left\\|\int_{0}^{h}\mathcal{T}(s)AU_{0}\,ds\right\\|_{X}\leq h\\|AU_{0}\\|_{X},$		(B.12)
	$\displaystyle\\|J_{3}\\|_{X}$	$\displaystyle\leq h\sup_{s\in[0,T]}\\|K(s;U(s))\\|_{X}.$		(B.13)

For $J_{2}$ , using the Lipschitz continuity of $a,b:[0,T]\to\mathbb{R}$ and $K(s:\cdot):X\to X$ , we can show

$\displaystyle\\|J_{2}\\|_{X}$	$\displaystyle\leq\int_{0}^{t}\\|K(t+h-s;U(t+h-s))-K(t+h-s;U(t-s))\\|_{X}\,ds$	(B.14)
	$\displaystyle\quad+\int_{0}^{t}\\|K(t+h-s;U(t-s))-K(t-s;U(t-s))\\|_{X}\,ds$	(B.15)
	$\displaystyle\leq C_{T_{0},M}h+C_{T_{0},M}\int_{0}^{t}\\|U(s+h)-U(s)\\|_{X}\,ds.$	(B.16)

Then, the Gronwall inequality implies

\|U(t+h)-U(t)\|_{X}\leq C_{T_{0},M}h,

that is, $U:[0,T]\to X$ is Lipschitz continuous. This further leads to

	$\displaystyle\\|K(t;U(t))-K(s;U(s))\\|_{X}$	$\displaystyle\leq\\|K(t;U(t))-K(s;U(t))\\|_{X}+\\|K(s;U(t))-K(s;U(s))\\|_{X}$		(B.17)
		$\displaystyle\leq C_{T_{0},M}\|t-s\|,$		(B.18)

i.e., $K(\cdot;U(\cdot)):[0,T]\to X$ is Lipschitz continuous, and hence, $K(\cdot;U(\cdot))\in W^{1,1}((0,T);X)$ . This enables us to apply [3, Lemma 4.16] and $u$ becomes a strong solution. This proves (v).

Next, we prove (vi). Let $T>0$ be arbitrary fixed, $I:=[0,T]$ , and

C_{T,a,b}:=\int_{0}^{T}(|a(s)|+|b(s)|)\,ds.

Let $\varepsilon>0$ be sufficiently small so that $2(1+C_{T,a,b})\varepsilon<1$ and let $\mathcal{B}_{\varepsilon}=\{U\in C([0,T];X);\,\sup_{t\in[0,T]}\|U(t)\|_{X}\leq 2(1+C_{T,a,b})\varepsilon\}$ . Define a map $\Phi:C(I:X)\to C(I;X)$ by

\Phi[U](t):=\mathcal{T}(t)U_{0}+\int_{0}^{t}\mathcal{T}(t-s)K(s;U(s))\,ds.

Then, for $U_{0}$ satisfying $\|U_{0}\|_{X}\leq\varepsilon$ and $U={}^{t}(u,v)\in\mathcal{B}_{\varepsilon}$ , we see that

$\displaystyle\\|\Phi[U](t)\\|_{X}$	$\displaystyle\leq\\|\mathcal{T}(t)U_{0}\\|_{X}+\int_{0}^{t}\\|\mathcal{T}(t-s)K(s;U(s))\\|_{X}\,ds$	(B.19)
	$\displaystyle\leq\\|U_{0}\\|_{X}+\int_{0}^{t}\left(\|b(s)\|\\|v(s)\\|_{H^{1}}+\|a(s)\|\\|\partial_{x}^{2}u(s)\\|_{H^{1}}\right)\,ds$	(B.20)
	$\displaystyle\quad+\int_{0}^{t}\\|\partial_{x}N(\partial_{x}u(s))\\|_{H^{1}}\,ds$	(B.21)
	$\displaystyle\leq(1+C_{T,a,b})\varepsilon+TC_{N}(2(1+C_{T,a,b})\varepsilon)^{2},$	(B.22)

where $C_{N}>0$ is a constant depending only on the nonlinearity $N$ . Similarly, we have, for $U,V\in\mathcal{B}_{\varepsilon}$ ,

	$\displaystyle\\|\Phi[U](t)-\Phi[V](t)\\|_{X}$	$\displaystyle\leq\int_{0}^{t}\\|K(s;U(s))-K(s;V(s))\\|_{X}\,ds$		(B.23)
		$\displaystyle\leq T\tilde{C}_{N}(2(1+C_{T,a,b})\varepsilon)\sup_{s\in[0,T]}\\|U(s)-V(s)\\|_{X},$		(B.24)

$\tilde{C}_{N}>0$ is a constant depending only on the nonlinearity $N$ . Therefore, taking $\varepsilon$ further small so that

TC_{N}(2(1+C_{T,a,b})\varepsilon)\leq 1,\quad T\tilde{C}_{N}(2(1+C_{T,a,b})\varepsilon)\leq\frac{1}{2},

we see that $\Phi$ is a contraction mapping on $\mathcal{B}_{\varepsilon}$ . This and the uniqueness of mild solution imply that the mild solution obtained in (i) can be extended to $[0,T]$ .

Finally, we prove (vii). Let $T>0$ , $I=[0,T]$ , $U_{0}\in Y$ , and let $U$ be the corresponding mild solution on $[0,T]$ to the initial data $U_{0}$ . We put $M:=\sup_{t\in I}\|U(t)\|_{X}$ . In order to justify the following energy method, we take a sequence $\{U_{0}^{(j)}\}_{j=1}^{\infty}$ from $[C_{0}^{\infty}(\mathbb{R})]^{2}$ such that $\lim_{j\to\infty}U_{0}^{(j)}=U_{0}$ in $X\cap Y$ . Then, the corresponding strong solution $U^{(j)}\in C(I;D(A))\cap C^{1}(I;X)$ to the data $U_{0}^{(j)}$ satisfies $\lim_{j\to\infty}U^{(j)}=U$ in $C(I;X)$ by the continuous dependence on the initial data. In particular, taking sufficiently large $j$ , we may suppose that $\sup_{j\in\mathbb{N},t\in I}\|U^{(j)}(t)\|_{X}\leq 2M$ .

Let

	$\displaystyle\chi\in C_{0}^{\infty}(\mathbb{R}),\quad 0\leq\chi\leq 1,\quad\chi(x)=\begin{cases}1&(\|x\|\leq 1),\\ 0&(\|x\|\geq 2),\end{cases}$
	$\displaystyle\chi_{n}(x):=\chi\left(\frac{x}{n}\right)\quad(n\in\mathbb{N}).$

By $\mathrm{supp}\,\chi_{n}\subset[-2n,2n]$ , we easily see that

$\displaystyle\|\partial_{x}(x^{2}\chi_{n}(x)^{2})\|$	$\displaystyle=\left\|2x\chi_{n}(x)^{2}+2\frac{x^{2}}{n}\chi^{\prime}\left(\frac{x}{n}\right)\chi_{n}(x)\right\|\leq C\|x\|\chi_{n}(x),$	(B.25)
$\displaystyle\|\partial_{x}^{2}(x^{2}\chi_{n}(x)^{2})\|$	$\displaystyle=\left\|2\chi_{n}(x)^{2}+4\frac{x}{n}\chi^{\prime}\left(\frac{x}{n}\right)\chi_{n}(x)+2\frac{x^{2}}{n^{2}}\left(\left(\chi^{\prime}\left(\frac{x}{n}\right)\right)^{2}+\chi^{\prime\prime}\left(\frac{x}{n}\right)\chi_{n}(x)\right)\right\|$	(B.26)
	$\displaystyle\leq C$	(B.27)

with some constant $C>0$ . Denote $U={}^{t}(u,\partial_{t}u)$ , $U^{(j)}={}^{t}(u^{(j)},\partial_{t}u^{(j)})$ , and consider

	$\displaystyle E_{n}(t;u)$	$\displaystyle:=\int_{\mathbb{R}}x^{2}\chi_{n}(x)^{2}\left(\|\partial_{t}u(t,x)\|^{2}+a(t)\|\partial_{x}u(x)\|^{2}+\|\partial_{x}^{2}u(t,x)\|^{2}+\|u(t,x)\|^{2}\right)\,dx,$
	$\displaystyle E(t;u)$	$\displaystyle:=\int_{\mathbb{R}}x^{2}\left(\|\partial_{t}u(t,x)\|^{2}+a(t)\|\partial_{x}u(x)\|^{2}+\|\partial_{x}^{2}u(t,x)\|^{2}+\|u(t,x)\|^{2}\right)\,dx.$

Note that $E_{n}(t;u^{(j)})$ is finite thanks to $\chi_{n}$ . Differentiating it, we have

	$\displaystyle\frac{d}{dt}E_{n}(t;u^{(j)})$	$\displaystyle=2\int_{\mathbb{R}}x^{2}\chi_{n}(x)^{2}\left(\partial_{t}u^{(j)}\partial_{t}^{2}u^{(j)}+a(t)\partial_{x}u^{(j)}\partial_{t}\partial_{x}u^{(j)}+\partial_{x}^{2}u^{(j)}\partial_{t}\partial_{x}^{2}u^{(j)}\right)\,dx$		(B.28)
		$\displaystyle\quad+2\int_{\mathbb{R}}x^{2}\chi_{n}(x)^{2}u^{(j)}\partial_{t}u^{(j)}\,dx+\int_{\mathbb{R}}x^{2}\chi_{n}(x)^{2}a^{\prime}(t)\|\partial_{x}u^{(j)}\|^{2}\,dx.$		(B.29)

By the integration by parts and using the equation (1.1), the right-hand side can be written as

	$\displaystyle 2\int_{\mathbb{R}}x^{2}\chi_{n}(x)^{2}\partial_{t}u^{(j)}\left(-b(t)\partial_{t}u^{(j)}+\partial_{x}N(\partial_{x}u^{(j)})\right)\,dx$		(B.30)
	$\displaystyle\quad-2\int_{\mathbb{R}}\partial_{x}(x^{2}\chi_{n}(x)^{2})a(t)\partial_{x}u^{(j)}\partial_{t}u^{(j)}\,dx$		(B.31)
	$\displaystyle\quad+4\int_{\mathbb{R}}\partial_{x}(x^{2}\chi_{n}(x)^{2})a(t)\partial_{x}^{3}u^{(j)}\partial_{t}u^{(j)}\,dx+2\int_{\mathbb{R}}\partial_{x}^{2}(x^{2}\chi_{n}(x)^{2})a(t)\partial_{x}^{2}u^{(j)}\partial_{t}u^{(j)}\,dx$		(B.32)
	$\displaystyle\quad+2\int_{\mathbb{R}}x^{2}\chi_{n}(x)^{2}u^{(j)}\partial_{t}u^{(j)}\,dx+\int_{\mathbb{R}}x^{2}\chi_{n}(x)^{2}a^{\prime}(t)\|\partial_{x}u^{(j)}\|^{2}\,dx.$		(B.33)

The above quantity can be further estimated by

C(2M)^{2}+C_{T,a,b,M}E_{n}(t;u^{(j)})

with some constants $C,C_{T,a,b,M}>0$ . Hence, the Gronwall inequality implies

E_{n}(t;u^{(j)})\leq\tilde{C}_{T,a,b,M},

where the constant $\tilde{C}_{T,a,b,M}$ is independent of $n$ and $j$ . Letting $j\to\infty$ first and using the continuous dependence on the initial data, we have

\int_{\mathbb{R}}x^{2}\chi_{n}(x)^{2}\left(|\partial_{t}u(t,x)|^{2}+a(t)|\partial_{x}u(x)|^{2}+|\partial_{x}^{2}u(t,x)|^{2}+|u(t,x)|^{2}\right)\,dx\leq\tilde{C}_{T,a,b,M}.

Then, letting $n\to\infty$ , we conclude

\int_{\mathbb{R}}x^{2}\left(|\partial_{t}u(t,x)|^{2}+a(t)|\partial_{x}u(x)|^{2}+|\partial_{x}^{2}u(t,x)|^{2}+|u(t,x)|^{2}\right)\,dx\leq\tilde{C}_{T,a,b,M},

which shows $U(t)\in Y$ for any $t\in[0,T]$ . The continuity of $\|U(t)\|_{Y}$ in $t$ follows from the estimate

\displaystyle|E_{n}(t;u^{(j)})-E_{n}(s;u^{(j)})|

\displaystyle\leq\int_{s}^{t}\left|\frac{d}{d\sigma}E_{n}(\sigma;u^{(j)})\right|\,d\sigma\leq C_{T,a,b,M}(t-s)

(B.34)

for $s<t$ and taking the limits $j\to\infty$ and $n\to\infty$ .

∎

Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers JP18H01132, JP20K14346.

References

[1] J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399–418.
[2] M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.
[3] Th. Cazenave, A. Haraux, An introduction to semilinear evolution equations, Oxford University Press, 1998.
[4] M. D’Abbicco, M.-R. Ebert, Asymptotic profiles and critical exponents for a semilinear damped plate equation with time-dependent coefficients, Asymptotic Analysis 123 (2021), 1–40.
[5] Th. Gallay, G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations, 150 (1998), 42–97.
[6] C. Lizama, M. Murillo-Arcila, On the dynamics of the damped extensible beam 1D-equation, J. Math. Anal. Appl., 522 (2023), Paper No. 126954, 10 pp.
[7] P. Marcati, K. Nishihara, The $L^{p}$ - $L^{q}$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445–469.
[8] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci., 12 (1976), 383–390.
[9] K. Nishihara, Asymptotic profile of solutions for 1-D wave equation with time-dependent damping and absorbing semilinear term, Asymptot. Anal., 71 (2011), 185–205.
[10] R. Racke, S. Yoshikawa, Singular limits in the Cauchy problem for the damped extensible beam equation, J. Differential Equations, 259 (2015), 1297–1322.
[11] T. Suzuki, S. Yoshikawa, Stability of the steady state for the Falk model system of shape memory alloys, Math. Methods Appl. Sci., 30 (2007), 2233–2245.
[12] H. Takeda, S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping II: asymptotic profiles, J. Differential Equations, 253 (2012), 3061–3080.
[13] Y. Wakasugi, Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients, J. Math. Anal. Appl., 447 (2017), 452–487.
[14] J. Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations, 222 (2006), 487–514.
[15] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations, 232 (2007), 74–103.
[16] S. Woinovsky-Krieger, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35–36.
[17] S. Yoshikawa, Y. Wakasugi, Classification of asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients: Effective damping case, J. Differential Equations 272 (2021), 938–957.