On Initial-Boundary Value Problems for Some One Dimensional Quasilinear Wave Equations: Global Existence, Scattering and Rigidity

Abstract

We consider the initial-boundary value problem on $\mathbb {R}^{+}\times \mathbb {R}^{+}$ for some one dimensional systems of quasilinear wave equations with null conditions. We first prove that for homogeneous Dirichlet boundary values and sufficiently small initial data, classical solutions always globally exist. Then we show that the global solution will scatter, i.e., it will converge to some solution of one dimensional linear wave equations as time tends to infinity, in the energy sense. Finally we prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically.

1 Introduction

For the Cauchy problem of one dimensional semilinear wave equations with null conditions, [12] proves the global existence of classical solutions with small initial data. This result strengthens a previous one in [15], which shows the global existence under some stronger null conditions. The result in [12] can be viewed as a one dimensional and semilinear analogue of Christodoulou and Klainerman’s pioneering works for the global existence of classical solutions for nonlinear wave equations with null conditions in three space dimensions [2, 6], and of Alinhac’s global existence result for the case of two space dimensions [1] (see also [4, 16] for some thorough studies in the 2-D case). The proofs in [1, 2, 6] are based on the decay mechanism in time of linear waves, however, in one space dimension waves do not decay in time. In the one dimensional case, the mechanism for the global existence is the interaction of waves with different speeds, which will lead to the decay in time of nonlinear terms. In order to display this mechanism, a kind of weighted energy estimates with positive weights is developed in [12].

Then for the Cauchy problem of one dimensional quasilinear wave equations with null conditions, Zha [17] shows the global existence of classical solutions in the small data setting. The proof in [17] is based on weighted energy estimates with positive weights in [12], some space-time weighted energies and new observations concerning the null structure in the quasilinear part.

Once the global solution is obtained, a further topic is its asymptotic behavior, including the scattering property and rigidity property. Zha [17] shows that the global solution will scatter, i.e., it will converge to some solution of one dimensional linear wave equations as time tends to infinity, in the unweighted energy sense. Then Li [7] proves that the global solution scatters in the weighted energy sense, and admits some rigidity property: if the scattering data vanish, then the global solution will also vanish identically. This is consistent with physical interpretations of scattering fields: the detecting fields of waves are the waves detected from a far-away observer. Therefore, the rigidity property has the following physical intuition: if no waves are detected by the far-away observers, then there are no waves at all emanating from the solution.

Based on the above global existence, scattering and rigidity results in the Cauchy problem case, in this paper, we intend to treat the corresponding topics in the initial-boundary value problem case.

The outline of this paper is as follows. The remainder of this introduction will be devoted to the description of a statement of main result and its applications. In Sect. 2, some necessary tools used to prove Theorem 1.1 are introduced. Section 3 is devoted to the proof of Theorem 1.1.

1.1 Main Result

Let (t, x) denote the usual Cartesian coordinates in ${\mathbb {R}}^{+}\times {\mathbb {R}}^{+}$, and define also the null coordinates

$$\begin{aligned} \xi =\frac{t+x}{2}, \quad \eta =\frac{t-x}{2}, \end{aligned}$$

(1.1)

as well as the corresponding null vector fields

$$\begin{aligned} \partial _{\xi }=\partial _t+\partial _x, \quad \partial _{\eta }=\partial _t-\partial _x. \end{aligned}$$

(1.2)

We also denote briefly $u_{\xi }=\partial _{\xi }u$ and $u_{\eta }=\partial _{\eta }u$.

Consider the following one dimensional system of quasilinear wave equations

$$\begin{aligned} u_{\xi \eta }&=A_1(u,u_{\xi },u_{\eta })u_{\xi \eta } +A_2(u,u_{\xi },u_{\eta })u_{\xi \xi }+A_3(u,u_{\xi },u_{\eta })u_{\eta \eta }+F(u,u_{\xi },u_{\eta }), \end{aligned}$$

(1.3)

where the unknown function $u=u(t,x): {\mathbb {R}}^{+}\times {\mathbb {R}}^{+}\longrightarrow {\mathbb {R}}^{n}$, for $i=1,2,3$, $A_i: {\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{n\times n}$ are given smooth and matrix valued functions, and $F: {\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{n}$ is a given smooth and vector valued function. Moreover, we will always assume that $A_i ~(i=1,2,3)$ are symmetric, which means that they take values of symmetric matrixes.

We call that the system (1.3) satisfies the null conditions, if near the origin in ${\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\times {\mathbb {R}}^{n}$, it holds that

$$\begin{aligned} A_1(u,u_{\xi },u_{\eta })&={\mathscr {O}}(|u|+|u_{\xi }|+|u_{\eta }|), \end{aligned}$$

(1.4)

$$\begin{aligned} A_2(u,u_{\xi },u_{\eta })&={\mathscr {O}}(|u_{\eta }|),\end{aligned}$$

(1.5)

$$\begin{aligned} A_3(u,u_{\xi },u_{\eta })&={\mathscr {O}}(|u_{\xi }|),\end{aligned}$$

(1.6)

$$\begin{aligned} F(u,u_{\xi },u_{\eta })&={\mathscr {O}}(|u_{\xi }||u_{\eta }|). \end{aligned}$$

(1.7)

The purpose of this paper is to treat initial-boundary value problem on ${\mathbb {R}}^{+}\times {\mathbb {R}}^{+}$ for one dimensional system of quasilinear wave equations (1.3). Consider the homogeneous boundary condition

$$\begin{aligned} u(t,0)=0,~t\ge 0, \end{aligned}$$

(1.8)

and the initial condition

$$\begin{aligned} t=0: u=u_0(x), \quad u_{t}=u_{1}(x), \quad x\ge 0. \end{aligned}$$

(1.9)

As usual, we also always assume that the initial data satisfy the compatibility conditions of order two. That means, $u_0(0)=0, u_1(0)=0$, $u'_0(0)=0$ and $u'_1(0)=0$.

Inspired by [7, 12, 15, 17] and we have the following

Conjecture 1.1

For the initial-boundary value problem (1.3), (1.8), (1.9), if $A_1, A_2, A_3$ are symmetric, the null conditions (1.4), (1.5), (1.6), (1.7) are satisfied, and smooth initial data admit suitable decay at infinity and are sufficiently small, then (1.3), (1.8), (1.9) admits a unique global classical solution, and the global solution scatters and admits the rigidity property.

For the semilinear case, the global existence and scattering parts of Conjecture 1.1 is verified in [19]. The purpose of this paper is to investigate the quasilinear case, which is more complicated than the semilinear one. The main difficulty for the initial-boundary value problem lies in that, in the (higher order) energy estimates, we can only use the time derivative, which preserves the homogeneous boundary conditions, as the commuting vector field. A consequence of this fact is that there will be some uncontrollable terms in the quasilinear part after integrating by parts argument, under the null conditions (1.5) and (1.6). Noting that in the Cauchy problem case [17], we can solve this problem by using the null coordinates $Z=\{\partial _{\xi },\partial _{\eta }\}$ as the commuting vector field.

As the first step, instead of (1.5) and (1.6), we introduce the following stronger conditions:

$$\begin{aligned} A_2(u,u_{\xi },u_{\eta })&={\mathscr {O}}(|u_{\eta }|^2), \end{aligned}$$

(1.5′)

$$\begin{aligned} A_3(u,u_{\xi },u_{\eta })&={\mathscr {O}}(|u_{\xi }|^2). \end{aligned}$$

(1.6′)

On the one hand, the change of the original form of the null condition from (1.5) and (1.6) to (1.5′) and (1.6′) is required in our later estimate (see Lemmas 2.5 and 2.7). On the other hand, we should point out that some physically meaningful one dimensional models which fall into the form of quasilinear wave equations (1.3), such as timelike minimal surface equation [10] and Faddeev model [3], indeed satisfy the stronger null conditions (1.4), (1.5′), (1.6′) and (1.7). See Sect. 1.2 for more details.

In order to describe the asymptotic behavior of the global solution precisely, now we introduce the concepts of scattering and rigidity. We say that a function $u=u(t,x)\in C({\mathbb {R}}^{+};{\dot{H}}^1({\mathbb {R}}^{+}))\cap C^1({\mathbb {R}}^{+};L^2({\mathbb {R}}^{+}))$ is asymptotically free in the energy sense, if there is $({\overline{u}}_0,{\overline{u}}_1)\in {\dot{H}}^1({\mathbb {R}}^{+})\times L^2({\mathbb {R}}^{+})$ such that

$$\begin{aligned} \lim _{t\rightarrow +\infty }\big (\Vert u_{\xi }-{\overline{u}}_{\xi }\Vert _{L^2_{x}({\mathbb {R}}^{+})}+\Vert u_{\eta }-{\overline{u}}_{\eta }\Vert _{L^2_{x}({\mathbb {R}}^{+})}\big )=0, \end{aligned}$$

(1.10)

where ${\overline{u}}\in C({\mathbb {R}}^{+};{\dot{H}}^1({\mathbb {R}}^{+}))\cap C^1({\mathbb {R}}^{+};L^2({\mathbb {R}}^{+}))$ is the unique global solution to the initial-boundary value problem of homogeneous linear wave equations

$$\begin{aligned} {\left\{ \begin{array}{ll} {\overline{u}}_{\xi \eta }=0, ~t>0,x>0,\\ {\overline{u}}(t,0)=0, ~~t\ge 0,\\ t=0: {\overline{u}}={\overline{u}}_0(x), {\overline{u}}_{t}={\overline{u}}_{1}(x),~ x\ge 0. \end{array}\right. } \end{aligned}$$

(1.11)

By definition, we say that a global solution u to the initial-boundary value problem (1.3), (1.8), (1.9) scatters, if it is asymptotically free. The corresponding initial data $({\overline{u}}_0,{\overline{u}}_1)$ is called the “scattering data”. We also say that the rigidity property holds, if the vanishing of scattering data implies the vanishing of global solution.

The main result of this paper is the following

Theorem 1.1

For the system (1.3), assume that $A_1, A_2, A_3$ are symmetric, (1.4), (1.5′), (1.6′), (1.7) hold. Then for all $0<\delta <1$, there exist a positive constant $\varepsilon _0$ such that for any $0<\varepsilon \le \varepsilon _0$, if

$$\begin{aligned} \sum _{l=0}^{3}\Vert \langle x\rangle ^{1+\delta }\partial _x^{l}u_0\Vert _{L_{x}^2({\mathbb {R}}^{+})}+\sum _{l=0}^{2}\Vert \langle x\rangle ^{1+\delta }\partial _x^{l}u_1\Vert _{L_{x}^2({\mathbb {R}}^{+})}\le \varepsilon , \end{aligned}$$

(1.12)

then the initial-boundary value problem (refquasiwave), (1.8), (1.9) admits a unique global classical solution u. Furthermore, the global solution will scatter, and the rigidity property holds.

Remark 1.1

The method of proof for Theorem 1.1 can also treat the initial-boundary value problem outside of a bounded internal, which is also called the exterior domain problem. In the 3-D case, the exterior domain problem analogue of Christodoulou and Klainerman’s global existence results for nonlinear wave equations with null conditions [2, 6] are treated in [5, 8, 13, 14], etc. While, in the 2-D case, how to get the exterior domain problem analogue of Alinhac’s global existence result [1] is still open until now.

Remark 1.2

For the Cauchy problem of quasilinear wave equations in higher space dimensions with small initial data, the scattering of classical solutions is studied systematically in [4]. But to the best of the authors’ knowledge, for the exterior domain problem of quasilinear wave equations, there is no such scattering result until now.

Remark 1.3

We point out that the rigidity part of Theorem  1.1 is inspired by [7, 9], in which the rigidity aspects of scattering problems are studied, for the Cauchy problems of MHD equations and one-dimension quasilinear wave equations with null conditions, respectively. We also point out that the approach in this paper is different from the corresponding ones in above two works. Specifically speaking, in order to treat the effect of the boundary, our main innovation is that on the base of the scattering result in the unweighetd energy sense, rigidity result follows directly from the energy estimate in Lemma  2.7, which relies on the time reversal invariance of the system (1.3). While [7, 9] need to introduce some spatial position parameters, which can not be used in the initial-boundary value problem case.

1.2 Applications

In this subsection, we will give two applications of Theorem 1.1.

We first apply Theorem 1.1 to the initial-boundary value problem for the equation of timelike minimal surface in Minkowski space ${\mathbb {R}}^{1+(1+n)}$ in the first quadrant. The equation describing graphical timelike minimal surface in ${\mathbb {R}}^{1+(1+n)}$ can be written as the following system for one-dimension quasilinear wave equations (see [10]):

$$\begin{aligned} (1+|u_x|^2)u_{tt}-2u_t\cdot u_x u_{tx}-(1-|u_t|^2)u_{xx}=0, \end{aligned}$$

(1.13)

where $u: {\mathbb {R}}^{+}\times {\mathbb {R}}^{+}\longrightarrow {\mathbb {R}}^n$, (t, x, u(t, x)) stands for a point on the graph of timelike minimal surface in ${\mathbb {R}}^{1+(1+n)}$. We first rewrite the timelike minimal surface equation (1.13) in the null coordinate $(\xi ,\eta )$ as follows

$$\begin{aligned} u_{\xi \eta }=\frac{1}{2}u_{\xi }\cdot u_{\eta }u_{\xi \eta }-\frac{1}{4}|u_{\eta }|^2u_{\xi \xi }-\frac{1}{4}|u_{\xi }|^2u_{\eta \eta }. \end{aligned}$$

(1.14)

It is obvious that (1.14) falls into the form of the system (1.3) with conditions (1.4), (1.5′), (1.6′), (1.7).

Consider the timelike minimal surface equation (1.14) on ${\mathbb {R}}^{+}\times {\mathbb {R}}^{+}$ with the homogeneous boundary condition

$$\begin{aligned} u(t,0)=0,~t\ge 0, \end{aligned}$$

(1.15)

and the initial condition

$$\begin{aligned} t=0: u=u_0(x),~ u_{t}=u_{1}(x), x\ge 0. \end{aligned}$$

(1.16)

Theorem 1.1 implies the following

Theorem 1.2

For all $0<\delta <1$, there exist a positive constant $\varepsilon _0$ such that for any $0<\varepsilon \le \varepsilon _0$, if

$$\begin{aligned} \sum _{l=0}^{3}\Vert \langle x\rangle ^{1+\delta }\partial _x^{l}u_0\Vert _{L_{x}^2({\mathbb {R}}^{+})}+\sum _{l=0}^{2}\Vert \langle x\rangle ^{1+\delta }\partial _x^{l}u_1\Vert _{L_{x}^2({\mathbb {R}}^{+})}\le \varepsilon , \end{aligned}$$

(1.17)

then the initial-boundary value problem for timelike minimal surface equation (1.14), (1.15), (1.16) admits a unique global classical solution u. Furthermore, the global solution will scatter, and the rigidity property holds.

We now turn to the Faddeev model, which is an important model that describes heavy elementary particles by knotted topological solitons in quantum field theory (see [3]). The original unknown function in this model is a map from the Minkowski space ${\mathbb {R}}^{1+d}$ to the unit sphere in ${\mathbb {R}}^{3}$, $\mathbf{{n}}: {\mathbb {R}}^{1+d} \longrightarrow {\mathbb {S}}^2$. Let

$$\begin{aligned} \mathbf{{n}}=(\cos u^{1}\cos u^2, \cos u^{1}\sin u^{2}, \sin u^{1})^{{\mathbb {T}}} \end{aligned}$$

(1.18)

be a vector in the unit sphere. Here $u^{1}: {\mathbb {R}}^{1+d} \longrightarrow [-\pi ,\pi ]$ and $u^{2}: {\mathbb {R}}^{1+d} \longrightarrow [-\frac{\pi }{2},\frac{\pi }{2}]$ stand for the latitude and longitude, respectively. The system for $u=(u^1,u^2)^{T}$ is (see [18])

$$\begin{aligned} {\left\{ \begin{array}{ll} \Box u^{1}=F(u^{1}, D u^{1}, Du^{2}, D^2u^{1}, D^2u^{2}),\\ \Box u^{2}=G(u^{1}, Du^{1}, Du^{2}, D^2u^{1}, D^2u^{2}), \end{array}\right. } \end{aligned}$$

(1.19)

where $\Box =\partial _t^2-\Delta$ is the wave operator on ${\mathbb {R}}^{1+d}$,

$$\begin{aligned}&F(u^{1}, D u^{1}, Du^{2}, D^2u^{1}, D^2u^{2})\nonumber \\&\quad =-\frac{1}{2}\sin (2u^{1})Q(u^{2},u^{2})-\frac{1}{4}\sin (2u^{1})Q_{\mu \nu }(u^{1},u^{2})Q^{\mu \nu }(u^{1},u^{2})\nonumber \\&\qquad -\frac{1}{2}\cos ^2u^{1} Q_{\mu \nu } \big (u^{2},Q^{\mu \nu }(u^{1},u^{2})\big ), \end{aligned}$$

(1.20)

$$\begin{aligned}&G(u^{1}, D u^{1}, Du^{2}, D^2u^{1}, D^2u^{2})\nonumber \\&\quad =\sin ^2u^{1} \Box u^{2}+\sin (2u^{1})Q(u^{1},u^{2})+\frac{1}{2}\cos ^2u^{1} Q_{\mu \nu } \big (u^{1},Q^{\mu \nu }(u^{1},u^{2})\big ), \end{aligned}$$

(1.21)

and the null forms

$$\begin{aligned} Q(f,g)&=\partial _tf\partial _tg-\nabla f\cdot \nabla g, ~~ Q_{\mu \nu }(f,g)=\partial _{\mu }f\partial _{\nu }g-\partial _{\nu }f\partial _{\mu }g,~\mu ,\nu \nonumber \\&=0,1,\cdots ,d. \end{aligned}$$

(1.22)

In the one dimensional case, i.e., $d=1$, noting that in the null coordinate $(\xi ,\eta )$,

$$\begin{aligned} Q(f,g)=\frac{1}{2}\big (f_{\xi }g_{\eta }+f_{\eta }g_{\xi }\big ),~~ Q_{01}(f,g)=-\frac{1}{2}\big (f_{\xi }g_{\eta }-f_{\eta }g_{\xi }\big ), \end{aligned}$$

(1.23)

by some computations, we can rewrite the Faddeev model as follows

$$\begin{aligned} u_{\xi \eta }&=A_1(u,u_{\xi },u_{\eta })u_{\xi \eta }+ +A_2(u,u_{\xi },u_{\eta })u_{\xi \xi }+A_3(u,u_{\xi },u_{\eta })u_{\eta \eta }+F(u,u_{\xi },u_{\eta }), \end{aligned}$$

(1.24)

where $u=(u^1,u^2),~ F=(F^1,F^2)$,

$$\begin{aligned}&A_1=\frac{1}{4}\cos ^2u^1 \begin{bmatrix} 2u^2_{\xi }u^2_{\eta }&-(u^1_{\xi }u^2_{\eta }+u^1_{\eta }u^2_{\xi })u^1_{\xi }u^2_{\eta }+u^1_{\eta }u^2_{\xi })&2u^1_{\xi }u^1_{\eta }+4\tan ^2u^1 \end{bmatrix}, \end{aligned}$$

(1.25)

$$\begin{aligned}&A_2=\frac{1}{4}\cos ^2u^1 \begin{bmatrix} -(u^2_{\eta })^2&{} u^1_{\eta }u^2_{\eta }\\ u^1_{\eta }u^2_{\eta }&{}-(u^1_{\eta })^2 \end{bmatrix}, A_3=\frac{1}{4}\cos ^2u^1 \begin{bmatrix} -(u^2_{\xi })^2&{} u^1_{\xi }u^2_{\xi }\\ u^1_{\xi }u^2_{\xi }&{}-(u^1_{\xi })^2 \end{bmatrix},\end{aligned}$$

(1.26)

$$\begin{aligned}&F^1=-\frac{1}{2}(\sin 2u^1)u^2_{\xi }u^2_{\eta }+\frac{1}{8}(\sin 2u^1)((u^1_{\xi })^2(u^2_{\eta })^2+(u^1_{\eta })^2(u^2_{\xi })^2-2u^1_{\xi }u^1_{\eta }u^2_{\xi }u^2_{\eta } ),\end{aligned}$$

(1.27)

$$\begin{aligned}&F^2= \frac{1}{2}(\sin 2u^1)(u^1_{\xi }u^2_{\eta }+u^1_{\eta }u^2_{\xi }). \end{aligned}$$

(1.28)

Thus, conditions (1.4), (1.5′), (1.6′) and (1.7) are satisfied.

Consider the one dimensional Faddeev model (1.24) on ${\mathbb {R}}^{+}\times {\mathbb {R}}^{+}$ with the homogeneous boundary condition

$$\begin{aligned} u(t,0)=0,~t\ge 0, \end{aligned}$$

(1.29)

and the initial condition

$$\begin{aligned} t=0: u=u_0(x),~ u_{t}=u_{1}(x), x\ge 0. \end{aligned}$$

(1.30)

Theorem 1.1 implies the following

Theorem 1.3

For all $0<\delta <1$, there exist a positive constant $\varepsilon _0$ such that for any $0<\varepsilon \le \varepsilon _0$, if

$$\begin{aligned} \sum _{l=0}^{3}\Vert \langle x\rangle ^{1+\delta }\partial _x^{l}u_0\Vert _{L_{x}^2({\mathbb {R}}^{+})}+\sum _{l=0}^{2}\Vert \langle x\rangle ^{1+\delta }\partial _x^{l}u_1\Vert _{L_{x}^2({\mathbb {R}}^{+})}\le \varepsilon , \end{aligned}$$

(1.31)

then the initial-boundary value problem for one dimensional Faddeev model (1.24), (1.29), (1.30) admits a unique global classical solution u. Furthermore, the global solution will scatter, and the rigidity property holds.

2 Preliminaries

2.1 Notations

For the convenience, we first introduce some notations. Fix $0<\delta <1$. Denote

$$\begin{aligned} \phi (x)=\langle x\rangle ^{2+2\delta }. \end{aligned}$$

(2.1)

It is easy to verify that

$$\begin{aligned} |\phi '(x)|\le 4\langle x\rangle ^{1+2\delta }. \end{aligned}$$

(2.2)

Denote

$$\begin{aligned} \psi (x)=\exp ({-\int _{-\infty }^{x}{\langle \rho \rangle ^{-(1+\delta )}}d\rho }). \end{aligned}$$

(2.3)

We can verify that

$$\begin{aligned} \psi '(x)=-\psi (x)\langle x\rangle ^{-(1+\delta )}. \end{aligned}$$

(2.4)

We note that there exists a positive constant c such that

$$\begin{aligned} c^{-1}\le \psi (x)\le c, \end{aligned}$$

(2.5)

thus

$$\begin{aligned} c^{-1}\langle x\rangle ^{-(1+\delta )}\le -\psi '(x)\le c\langle x\rangle ^{-(1+\delta )}. \end{aligned}$$

(2.6)

Now we will introduce some notations for energies. Denote the vector field $Z=\{\partial _{\xi },\partial _{\eta }\}$. For multi-index $a=(a_1,a_2)$, denote $Z^{a}=\partial _{\xi }^{a_1}\partial _{\eta }^{a_2}$ and $|a|=a_1+a_2$. Following [12], we will use the following weighted energy with positive weights

$$\begin{aligned} E(u(t))=\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert \langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2 \end{aligned}$$

(2.7)

and higher order energies

$$\begin{aligned} E_k(u(t))=\sum _{|a|\le k-1}E(Z^{a}u(t)). \end{aligned}$$

(2.8)

Inspired by [1] and [11], as in [17], denoting $S_t=[0,t]\times {\mathbb {R}}^+$, we further introduce the following space-time weighted energy

$$\begin{aligned} {\mathcal {E}}(u(t))=\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }\Vert ^2_{L^2_{s,x}(S_t)}+\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }\Vert ^2_{L^2_{s,x}(S_t)} \end{aligned}$$

(2.9)

and higher order energies

$$\begin{aligned} {\mathcal {E}}_k(u(t))=\sum _{|a|\le k-1}{\mathcal {E}}(Z^{a}u(t)). \end{aligned}$$

(2.10)

We also use notations

$$\begin{aligned} {\overline{E}}_3(u(t))&=\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert \langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2\nonumber \\&\quad +\Vert \langle \xi \rangle ^{1+\delta }u_{\xi \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert \langle \eta \rangle ^{1+\delta }u_{\eta \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2\nonumber \\&\quad \Vert \langle \xi \rangle ^{1+\delta }u_{\xi \xi \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert \langle \eta \rangle ^{1+\delta }u_{\eta \eta \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2 \end{aligned}$$

(2.11)

and

$$\begin{aligned} \overline{\overline{{E}}}_3(u(t))={E_3}(u(t))-{\overline{E}}_3(u(t)). \end{aligned}$$

(2.12)

Correspondingly, for the space-time weighted energy, we denote

$$\begin{aligned} \overline{{\mathcal {E}}}_3(u(t))&=\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }\Vert ^2_{L^2_{s,x}(S_t)}+\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }\Vert ^2_{L^2_{s,x}(S_t)}\nonumber \\&\quad +\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi \xi }\Vert ^2_{L^2_{s,x}(S_t)}+\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta \eta }\Vert ^2_{L^2_{s,x}(S_t)}\nonumber \\&\quad +\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi \xi \xi }\Vert ^2_{L^2_{s,x}(S_t)}+\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta \eta \eta }\Vert ^2_{L^2_{s,x}(S_t)} \end{aligned}$$

(2.13)

and

$$\begin{aligned} \overline{\overline{{{\mathcal {E}}}}}_3(u(t))={{\mathcal {E}}_3}(u(t))-\overline{{\mathcal {E}}}_3(u(t)). \end{aligned}$$

(2.14)

In the energy estimates, we can only use the time derivative as the commuting vector field. Thus we further introduce

$$\begin{aligned} \widetilde{{E}}_k(u(t))=\sum _{l=0}^{k-1}{E}(\partial _t^{l}u(t)), \end{aligned}$$

(2.15)

and

$$\begin{aligned} \widetilde{{\mathcal {E}}}_k(u(t))=\sum _{l=0}^{k-1}{{\mathcal {E}}}(\partial _t^{l}u(t)). \end{aligned}$$

(2.16)

We also denote the general energy and the corresponding space-time weighted energy by

$$\begin{aligned} E^{c}(u(t))=\Vert u_{\xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert u_{\eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2 \end{aligned}$$

(2.17)

and

$$\begin{aligned} {\mathcal {E}}^{c}(u(t))=\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}u_{\xi }\Vert ^2_{L^2_{s,x}(S_t)}+\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}u_{\eta }\Vert ^2_{L^2_{s,x}(S_t)} \end{aligned}$$

(2.18)

respectively.

2.2 Some Estimates

By the fundamental theorem of calculus, chain rule and Leibniz’s rule, it is easy to get the following two lemmas.

Lemma 2.1

Assume that u is smooth, $A_1=A_1(u,u_{\xi },u_{\eta }), A_2=A_2(u,u_{\xi },u_{\eta })$, $A_3=A_3(u,u_{\xi },u_{\eta })$ and $F=F(u,u_{\xi },u_{\eta })$ satisfies (1.4), (1.5′), (1.6′) and (1.7), respectively, and

$$\begin{aligned} |u|+|u_{\xi }|+|u_{\eta }|+|u_{\xi \xi }|+|u_{\eta \eta }|+|u_{\xi \eta }|\le \nu _{0}. \end{aligned}$$

(2.19)

Then we have

$$\begin{aligned} |\partial _{\xi }A_1|&\le C(|u_{\xi }|+|Zu_{\xi }|),\quad |\partial _{\eta }A_1|\le C(|u_{\eta }|+|Zu_{\eta }|), \end{aligned}$$

(2.20)

$$\begin{aligned} |\partial _{\xi }A_2|&\le C|u_{\eta }|(|u_{\xi }|+|Zu_{\xi }|),\quad |\partial _{\eta }A_2|\le C|u_{\eta }|(|u_{\eta }|+|Zu_{\eta }|),\end{aligned}$$

(2.21)

$$\begin{aligned} |\partial _{\xi }A_3|&\le C|u_{\xi }|(|u_{\xi }|+|Zu_{\xi }|), \quad |\partial _{\eta }A_3|\le C|u_{\xi }|(|u_{\eta }|+|Zu_{\eta }|), \end{aligned}$$

(2.22)

where $C=C(\nu _0)$ is a constant depending on $\nu _0$.

Lemma 2.2

Assume that u is smooth, $A_1=A_1(u,u_{\xi },u_{\eta }), A_2=A_2(u,u_{\xi },u_{\eta })$, $A_3=A_3(u,u_{\xi },u_{\eta })$ and $F=F(u,u_{\xi },u_{\eta })$ satisfies (1.4), (1.5′), (1.6′) and (1.7), respectively, and

$$\begin{aligned} |u|+|u_{\xi }|+|u_{\eta }|+|u_{\xi \xi }|+|u_{\eta \eta }|+|u_{\xi \eta }|\le \nu _{0}. \end{aligned}$$

(2.23)

Then we have

$$\begin{aligned} |\partial _{t}A_1|&\le C(|u_{t}|+|u_{t\xi }|+|u_{t\eta }|),\end{aligned}$$

(2.24)

$$\begin{aligned} |\partial ^2_{t}A_1|&\le C(|u_{t}|+|u_{tt}|+|u_{t\xi }|+|u_{t\eta }|+|u_{tt\xi }|+|u_{tt\eta }|),\end{aligned}$$

(2.25)

$$\begin{aligned} |\partial _{t}A_2|&\le C|u_{\eta }|,~~ |\partial ^2_{t}A_2|\le C(|u_{t\eta }|+|u_{tt\eta }|+|u_{\eta }||u_{tt\xi }|),\end{aligned}$$

(2.26)

$$\begin{aligned} |\partial _{t}A_3|&\le C|u_{\xi }|,~~ |\partial ^2_{t}A_3|\le C(|u_{t\xi }|+|u_{tt\xi }|+|u_{\xi }||u_{tt\eta }|),\end{aligned}$$

(2.27)

$$\begin{aligned} |\partial _{t}F|&\le C(|u_{\xi }||u_{t\eta }|+|u_{\eta }||u_{t\xi }|+|u_{\xi }||u_{\eta }|),\end{aligned}$$

(2.28)

$$\begin{aligned} |\partial ^2_{t}F|&\le C(|u_{\eta }||u_{tt\xi }|+|u_{\xi }||u_{tt\eta }|+|u_{t\xi }||u_{t\eta }|+|u_{\xi }||u_{t\eta }|+|u_{\eta }||u_{t\xi }|+|u_{\xi }||u_{\eta }|), \end{aligned}$$

(2.29)

where $C=C(\nu _0)$ is a constant depending on $\nu _0$.

The following pointwise estimates will be used frequently in the sequel.

Lemma 2.3

Let u be a smooth function on ${\mathbb {R}}^{+}\times {\mathbb {R}}^{+}$ with sufficient decay at spatial infinity.

Then it holds that

$$\begin{aligned}&\Vert u(t,\cdot )\Vert _{L^{\infty }({\mathbb {R}}^{+})}\le C E_3^{1/2}(u(t)), \end{aligned}$$

(2.30)

$$\begin{aligned}&\Vert \langle \xi \rangle ^{1+\delta }(|u_{\xi }|+|Zu_{\xi }|)\Vert _{L_{x}^{{\infty }}({\mathbb {R}}^{+})} +\Vert \langle \eta \rangle ^{1+\delta }(|u_{\eta }|+|Zu_{\eta }|)\Vert _{L_{x}^{{\infty }}({\mathbb {R}}^{+})}\nonumber \\&\quad \le C E_3^{1/2}(u(t)), \end{aligned}$$

(2.31)

and

$$\begin{aligned}&\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }(|u_{\xi }|+|Zu_{\xi }|)\Vert _{L^2_{s}L_{x}^{{\infty }}}+\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }(|u_{\eta }|+|Zu_{\eta }|)\Vert _{L^2_{s}L_{x}^{{\infty }}}\nonumber \\&\quad \le C {\mathcal {E}}_3^{1/2}(u(t)). \end{aligned}$$

(2.32)

Proof

It follows from the fundamental theorem of calculus and Hölder inequality that

$$\begin{aligned}&\Vert u(t,\cdot )\Vert _{L^{\infty }({\mathbb {R}}^{+})}\le \Vert u_x(t,\cdot )\Vert _{L_{x}^{1}({\mathbb {R}}^{+})}\le \Vert u_{\xi }\Vert _{L_{x}^{1}({\mathbb {R}}^{+})}+\Vert u_{\eta }\Vert _{L_{x}^{1}({\mathbb {R}}^{+})}\nonumber \\&\quad \le \Vert \langle \xi \rangle ^{-1-\delta }\Vert _{L_{x}^{2}({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^{2}({\mathbb {R}}^{+})}+\Vert \langle \eta \rangle ^{-1-\delta }\Vert _{L_{x}^{2}({\mathbb {R}}^{+})}\Vert \langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L_{x}^{2}({\mathbb {R}}^{+})}\nonumber \\&\quad \le C\big (\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^{2}({\mathbb {R}}^{+})}+\Vert \langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L_{x}^{2}({\mathbb {R}}^{+})}\big )\le C E^{1/2}(u(t))\le C E_3^{1/2}(u(t)). \end{aligned}$$

(2.33)

While (2.31) and (2.32) can be proved by Sobolev embedding $H^1({\mathbb {R}}^{+})\hookrightarrow L^{\infty }({\mathbb {R}}^{+})$ and the following fact

$$\begin{aligned} |\partial _{x}\langle \xi \rangle ^{1+\delta }|&\le C\langle \xi \rangle ^{1+\delta },~ |\partial _{x}(\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta })|\le C \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }, \end{aligned}$$

(2.34)

$$\begin{aligned} |\partial _{x}\langle \eta \rangle ^{1+\delta }|&\le C\langle \eta \rangle ^{1+\delta },~ |\partial _{x}(\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta })|\le C \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }. \end{aligned}$$

(2.35)

$\square$

The following lemma clarifies the relationship between various energies given in the above subsection.

Lemma 2.4

We have

$$\begin{aligned} \sup _{0\le s\le t}{\overline{E}}_3(u(s))\le C\big (\sup _{0\le s\le t}\overline{{\overline{E}}}_3(u(s))+\sup _{0\le s\le t}{\widetilde{E}}_3(u(s))\big ) \end{aligned}$$

(2.36)

and

$$\begin{aligned} \overline{{\mathcal {E}}}_3(u(t))\le C\big (\overline{\overline{{\mathcal {E}}}}_3(u(t))+\widetilde{{\mathcal {E}}}_3(u(t))\big ). \end{aligned}$$

(2.37)

Furthermore, if u satisfies (1.3), $A_1, A_2$, $A_3$ and F satisfies (1.4), (1.5′), (1.6′) and (1.7), respectively, and

$$\begin{aligned} \varepsilon _1=\sup _{0\le s\le t}E_3^{1/2}(u(s)) \end{aligned}$$

(2.38)

is sufficiently small. Then we also have

$$\begin{aligned} \sup _{0\le s\le t}\overline{{\overline{E}}}_3(u(s))\le C\sup _{0\le s\le t}{E}_3^2(u(t)) \end{aligned}$$

(2.39)

and

$$\begin{aligned} \overline{\overline{{\mathcal {E}}}}_3(u(t))\le C\sup _{0\le s\le t}{E}_3(u(s)){{\mathcal {E}}}_3(u(t)). \end{aligned}$$

(2.40)

Proof

In view of the definitions of energies, (2.36) and (2.37) just come from the following pointwise estimates

$$\begin{aligned} |u_{\xi \xi }|&\le 2|u_{t\xi }|+|u_{\eta \xi }|, \end{aligned}$$

(2.41)

$$\begin{aligned} |u_{\eta \eta }|&\le 2|u_{t\eta }|+|u_{\xi \eta }|, \end{aligned}$$

(2.42)

$$\begin{aligned} |u_{\xi \xi \xi }|&\le 2|u_{t\xi \xi }|+|u_{\eta \xi \xi }|\le 4|u_{tt\xi }|+2|u_{t\eta \xi }|+|u_{\eta \xi \xi }|\le 4|u_{tt\xi }|+3|u_{\eta \xi \xi }|+2|u_{\eta \eta \xi }| \end{aligned}$$

(2.43)

and

$$\begin{aligned} |u_{\eta \eta \eta }|&\le 2|u_{t\eta \eta }|+|u_{\xi \eta \eta }|\le 4|u_{tt\eta }|+2|u_{t\xi \eta }|+|u_{\xi \eta \eta }|\nonumber \\&\le 4|u_{tt\eta }|+3|u_{\xi \eta \eta }|+2|u_{\xi \xi \eta }|. \end{aligned}$$

(2.44)

The proof of (2.39) and (2.40) is similar with the corresponding ones in [17] (Lemma 2.4). We only give the sketch of the proof for (2.39). (2.40) can be proved similarly. We have

$$\begin{aligned} \overline{{\overline{E}}}_3(u(t))&=\Vert \langle \xi \rangle ^{1+\delta }u_{\eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert \langle \eta \rangle ^{1+\delta }u_{\xi \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert \langle \xi \rangle ^{1+\delta }u_{\eta \eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2\nonumber \\&\quad +\Vert \langle \xi \rangle ^{1+\delta }u_{\xi \eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert \langle \eta \rangle ^{1+\delta }u_{\xi \xi \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2+\Vert \langle \eta \rangle ^{1+\delta }u_{\xi \eta \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}^2. \end{aligned}$$

(2.45)

From the system (1.3), we have

$$\begin{aligned} u_{\eta \eta \xi }&=A_1u_{\eta \eta \xi } +A_2u_{\xi \eta \xi }+A_3u_{\eta \eta \eta }+\partial _{\eta }A_1u_{\eta \xi } +\partial _{\eta }A_2u_{\xi \xi }+\partial _{\eta }A_3u_{\eta \eta }+\partial _{\eta }F \end{aligned}$$

(2.46)

It follows from (2.46), (1.4), (1.5′), (1.6′), (1.7), Lemmas 2.1 and 2.3 that

$$\begin{aligned}&\Vert \langle \xi \rangle ^{1+\delta }u_{\eta \eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\quad \le C\Vert A_1\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\eta \eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}+C\Vert A_2\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\xi \eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{1+\delta }A_3\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert u_{\eta \eta \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}+C\Vert \partial _{\eta }A_1\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\qquad +C\Vert \partial _{\eta }A_2\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\xi \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})} +C\Vert \langle \xi \rangle ^{1+\delta }\partial _{\eta }A_3\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert u_{\eta \eta }\Vert _{L_{x}^{2}({\mathbb {R}}^{+})}\nonumber \\&\qquad +\Vert \langle \xi \rangle ^{1+\delta }\partial _{\eta }F\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\quad \le C\Vert |u|+|u_{\xi }|+|u_{\eta }|\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\eta \eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\qquad +C\Vert u_{\eta }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\xi \eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert u_{\eta \eta \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}+C\Vert |u_{\eta }|+|u_{\xi \eta }|\nonumber \\&\qquad +|u_{\eta \eta }|\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\qquad +C\Vert u_{\eta }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\xi \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})} +C\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert u_{\eta \eta }\Vert _{L_{x}^{2}({\mathbb {R}}^{+})} \nonumber \\&\qquad +C\Vert u_{\eta }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\eta \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})} +C\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert u_{\eta \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\qquad +C\Vert u_{\eta }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\le CE_3(u(t)). \end{aligned}$$

(2.47)

Other terms on the right hand side of (2.45) can be estimated similarly. $\square$

Remark 2.1

Some analogues of (2.39) and (2.40) are shown in the Cauchy problem case [17]. The simple and new estimates (2.36) and (2.37) will play key roles in the initial-boundary value problem case.

The following lemma is the key point in the proof of global existence part of Theorem 1.1.

Lemma 2.5

Assume that $v: {\mathbb {R}}^{+}\times {\mathbb {R}}^{+}\longrightarrow {\mathbb {R}}^n$ satisfies the following system of linear wave equations

$$\begin{aligned} v_{\xi \eta }&=A_1(u,u_{\xi },u_{\eta })v_{\xi \eta } +A_2(u,u_{\xi },u_{\eta })v_{\xi \xi }+A_3(u,u_{\xi },u_{\eta })v_{\eta \eta }+G, \end{aligned}$$

(2.48)

and homogeneous boundary condition $v(t,0)=0$. Here $A_1,A_2$, $A_3$ are symmetric and satisfies (1.4), (1.5′), (1.6′), respectively, u and $G: {\mathbb {R}}^{+}\times {\mathbb {R}}^{+}\longrightarrow {\mathbb {R}}^n$ are some given vector valued functions of (t, x), and

$$\begin{aligned} \varepsilon _1=\sup _{0\le s\le t}E_3(u(s))+{\mathcal {E}}_3(u(t)) \end{aligned}$$

(2.49)

is sufficiently small. Then it holds that

$$\begin{aligned}&\sup _{0\le s\le t}{E}(v(s))+ {{\mathcal {E}}}(v(t))\nonumber \\&\quad \le C E(v(0))+C\int _0^{t}\Vert \langle \xi \rangle ^{2+2\delta }v^{T}_{\xi }G\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds+C\int _0^{t}\Vert \langle \eta \rangle ^{2+2\delta }v^{T}_{\eta } G\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds. \end{aligned}$$

(2.50)

Proof

Multiply $2\psi (\eta )\phi (\xi )v^{{T}}_{\xi }$ on both sides of (2.48). Noting the symmetry of $A_1, A_2$ and $A_3$, by Leibniz’s rule we have

$$\begin{aligned}&\big (\psi (\eta )\phi (\xi )|v_{\xi }|^2\big )_{\eta }-\psi '(\eta )\phi (\xi )|v_{\xi }|^2\nonumber \\&\quad =\big (\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_1v_{\xi }\big )_{\eta }-\psi '(\eta )\phi (\xi )v_{\xi }^{{T}}A_1v_{\xi } -\psi (\eta )\phi (\xi )v_{\xi }^{{T}}\partial _{\eta }A_1v_{\xi }\nonumber \\&\qquad +\big (\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_2v_{\xi }\big )_{\xi }-\psi (\eta )\phi '(\xi )v_{\xi }^{{T}}A_2v_{\xi } -\psi (\eta )\phi (\xi )v_{\xi }^{{T}}\partial _{\xi }A_2v_{\xi }\nonumber \\&\qquad +\big (2\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_3v_{\eta }\big )_{\eta }-2\psi '(\eta )\phi (\xi )v_{\xi }^{{T}}A_3v_{\eta } -2\psi (\eta )\phi (\xi )v_{\xi }^{{T}}\partial _{\eta }A_3v_{\eta }\nonumber \\&\qquad -\big (\psi (\eta )\phi (\xi )v_{\eta }^{{T}}A_3v_{\eta }\big )_{\xi }+\psi (\eta )\phi '(\xi )v_{\eta }^{{T}}A_3v_{\eta } +\psi (\eta )\phi (\xi )v_{\eta }^{{T}}\partial _{\xi }A_3v_{\eta }\nonumber \\&\qquad +2\psi (\eta )\phi (\xi )v^{{T}}_{\xi }G. \end{aligned}$$

(2.51)

Similarly, multiply $2\psi (\xi )\phi (\eta )v^{{T}}_{\eta }$ on both sides of (2.48). The symmetry of $A_1, A_2$ and $A_3$ and Leibniz’s rule also imply

$$\begin{aligned}&\big (\psi (\xi )\phi (\eta )|v_{\eta }|^2\big )_{\xi }-\psi '(\xi )\phi (\eta )|v_{\eta }|^2\nonumber \\&\quad =\big (\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_1v_{\eta }\big )_{\xi }-\psi '(\xi )\phi (\eta )v_{\eta }^{{T}}A_1v_{\eta } -\psi (\xi )\phi (\eta )v_{\eta }^{{T}}\partial _{\xi }A_1v_{\eta }\nonumber \\&\qquad +\big (2\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_2v_{\xi }\big )_{\xi }-2\psi '(\xi )\phi (\eta )v_{\eta }^{{T}}A_2v_{\xi } -2\psi (\xi )\phi (\eta )v_{\eta }^{{T}}\partial _{\xi }A_2v_{\xi }\nonumber \\&\qquad -\big (\psi (\xi )\phi (\eta )v_{\xi }^{{T}}A_2v_{\xi }\big )_{\eta }+\psi (\xi )\phi '(\eta )v_{\xi }^{{T}}A_2v_{\xi } +\psi (\xi )\phi (\eta )v_{\xi }^{{T}}\partial _{\eta }A_2v_{\xi }\nonumber \\&\qquad +\big (\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_3v_{\eta }\big )_{\eta }-\psi (\xi )\phi '(\eta )v_{\eta }^{{T}}A_3v_{\eta } -\psi (\xi )\phi (\eta )v_{\eta }^{{T}}\partial _{\eta }A_3v_{\eta }\nonumber \\&\qquad +2\psi (\xi )\phi (\eta )v^{{T}}_{\eta }G. \end{aligned}$$

(2.52)

Integrating on $S_t=[0,t]\times {\mathbb {R}}^{+}$ on both sides of (2.51) and (2.52), by the fundamental theorem of calculus we have

$$\begin{aligned}&\int _{{\mathbb {R}}^{+}}\big (e(t,x)+{\widetilde{e}}(t,x)\big )dx+\int _0^{t}\big (p(s,0)+{\widetilde{p}}(s,0)\big )ds+\int _0^{t}\!\!\int _{{\mathbb {R}}^{+}}q(s,x)dxds\nonumber \\&\quad =\int _{{\mathbb {R}}^{+}}\big (e(0,x)+{\widetilde{e}}(0,x)\big )dx+\int _0^{t}\!\!\int _{{\mathbb {R}}^{+}}w(s,x)dxds\nonumber \\&\qquad +2\int _0^{t}\!\!\int _{{\mathbb {R}}^{+}}\psi (\eta )\phi (\xi )v^{{T}}_{\xi }Gdxds+2\int _0^{t}\!\!\int _{{\mathbb {R}}^{+}}\psi (\xi )\phi (\eta )v^{{T}}_{\eta }Gdxds, \end{aligned}$$

(2.53)

where

$$\begin{aligned} e&=\psi (\eta )\phi (\xi )|v_{\xi }|^2+\psi (\xi )\phi (\eta )|v_{\eta }|^2, \end{aligned}$$

(2.54)

$$\begin{aligned} {\widetilde{e}}&=-\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_1v_{\xi }-\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_2v_{\xi } -2\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_3v_{\eta }\nonumber \\&\quad +{{\psi (\eta )\phi (\xi )v_{\eta }^{{T}}A_3v_{\eta }}}\nonumber \\&\quad -\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_1v_{\eta } -2\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_2v_{\xi } +{{\psi (\xi )\phi (\eta )v_{\xi }^{{T}}A_2v_{\xi }}}\nonumber \\&\quad -\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_3v_{\eta }, \end{aligned}$$

(2.55)

$$\begin{aligned} p&=\psi (\eta )\phi (\xi )|v_{\xi }|^2-\psi (\xi )\phi (\eta )|v_{\eta }|^2, \end{aligned}$$

(2.56)

$$\begin{aligned} {\widetilde{p}}&=-\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_1v_{\xi }+\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_2v_{\xi } -2\psi (\eta )\phi (\xi )v_{\xi }^{{T}}A_3v_{\eta }\nonumber \\&\quad -\psi (\eta )\phi (\xi )v_{\eta }^{{T}}A_3v_{\eta }\nonumber \\&\quad +\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_1v_{\eta } +2\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_2v_{\xi } +\psi (\xi )\phi (\eta )v_{\xi }^{{T}}A_2v_{\xi }\nonumber \\&\quad -\psi (\xi )\phi (\eta )v_{\eta }^{{T}}A_3v_{\eta }, \end{aligned}$$

(2.57)

$$\begin{aligned} q&=-\psi '(\eta )\phi (\xi )|v_{\xi }|^2-\psi '(\xi )\phi (\eta )|v_{\eta }|^2, \end{aligned}$$

(2.58)

and

$$\begin{aligned} w&=-\psi '(\eta )\phi (\xi )v_{\xi }^{{T}}A_1v_{\xi }-2\psi '(\eta )\phi (\xi )v_{\xi }^{{T}}A_3v_{\eta } -\psi (\eta )\phi '(\xi )v_{\xi }^{{T}}A_2v_{\xi }\nonumber \\&\quad +{{\psi (\eta )\phi '(\xi )v_{\eta }^{{T}}A_3v_{\eta }}} \nonumber \\&\quad -\psi '(\xi )\phi (\eta )v_{\eta }^{{T}}A_1v_{\eta }-2\psi '(\xi )\phi (\eta )v_{\eta }^{{T}}A_2v_{\xi }\nonumber \\&\quad +{{\psi (\xi )\phi '(\eta )v_{\xi }^{{T}}A_2v_{\xi }}} -\psi (\xi )\phi '(\eta )v_{\eta }^{{T}}A_3v_{\eta }\nonumber \\&\quad -\psi (\eta )\phi (\xi )v_{\xi }^{{T}}\partial _{\eta }A_1v_{\xi }\nonumber \\&\quad -\psi (\eta )\phi (\xi )v_{\xi }^{{T}}\partial _{\xi }A_2v_{\xi } -2\psi (\eta )\phi (\xi )v_{\xi }^{{T}}\partial _{\eta }A_3v_{\eta } +{{\psi (\eta )\phi (\xi )v_{\eta }^{{T}}\partial _{\xi }A_3v_{\eta }}} \nonumber \\&\quad -\psi (\xi )\phi (\eta )v_{\eta }^{{T}}\partial _{\xi }A_1v_{\eta } -2\psi (\xi )\phi (\eta )v_{\eta }^{{T}}\partial _{\xi }A_2v_{\xi } \nonumber \\&\quad +{{\psi (\xi )\phi (\eta )v_{\xi }^{{T}}\partial _{\eta }A_2v_{\xi }}} -\psi (\xi )\phi (\eta )v_{\eta }^{{T}}\partial _{\eta }A_3v_{\eta }. \end{aligned}$$

(2.59)

In view of (2.1) and (2.5), we have

$$\begin{aligned} c^{-1}e(t,x)\le |\langle \xi \rangle ^{1+\delta }v_{\xi }|^2+|\langle \eta \rangle ^{1+\delta }v_{\eta }|^2\le ce(t,x) \end{aligned}$$

(2.60)

It follows from (2.1), (2.5), (1.4), (1.5′), (1.6′) and Lemma 2.3 that

$$\begin{aligned} |{\widetilde{e}}(t,x)|&\le C\langle \xi \rangle ^{2+2\delta }|v_{\xi }^{{T}}A_1v_{\xi }|+C\langle \xi \rangle ^{2+2\delta }|v_{\xi }^{{T}}A_2v_{\xi }|\nonumber \\&\quad +C\langle \xi \rangle ^{2+2\delta }|v_{\xi }^{{T}}A_3v_{\eta }|+C\langle \xi \rangle ^{2+2\delta }|v_{\eta }^{{T}}A_3v_{\eta }|\nonumber \\&\quad +C\langle \eta \rangle ^{2+2\delta }|v_{\eta }^{{T}}A_1v_{\eta }| +C\langle \eta \rangle ^{2+2\delta }|v_{\eta }^{{T}}A_2v_{\xi }| +C\langle \eta \rangle ^{2+2\delta }|v_{\xi }^{{T}}A_2v_{\xi }|\nonumber \\&\quad +C\langle \eta \rangle ^{2+2\delta }|v_{\eta }^{{T}}A_3v_{\eta }|\nonumber \\&\le C|\langle \xi \rangle ^{1+\delta }v_{\xi }|^2(|u|+|u_{\xi }|+|u_{\eta }|)+ C|\langle \xi \rangle ^{1+\delta }v_{\xi }|^2||u_{\eta }|^2\nonumber \\&\quad +C|\langle \xi \rangle ^{1+\delta }v_{\xi }||\langle \xi \rangle ^{1+\delta }u_{\xi }||u_{\xi }v_{\eta }|+{{C|\langle \xi \rangle ^{1+\delta }u_{\xi }|^2|v_{\eta }|^2}} \nonumber \\&\quad +C|\langle \eta \rangle ^{1+\delta }v_{\eta }|^2(|u|+|u_{\xi }|+|u_{\eta }|) +C|\langle \eta \rangle ^{1+\delta }v_{\eta }|\langle \eta \rangle ^{1+\delta }u_{\eta }|u_{\eta }v_{\xi }|\nonumber \\&\quad +{{C\langle \eta \rangle ^{1+\delta }u_{\eta }|^2|v_{\xi }|^2}} +C|\langle \eta \rangle ^{1+\delta }v_{\eta }|^2|u_{\xi }|^2\nonumber \\&\quad \le C\big (E_3^{1/2}(u(t))+E_3(u(t))\big ) \big (|\langle \xi \rangle ^{1+\delta }v_{\xi }|^2+ |\langle \eta \rangle ^{1+\delta }v_{\eta }|^2\big ). \end{aligned}$$

(2.61)

Noting (2.49), if $\varepsilon _1$ is sufficiently small, we can get

$$\begin{aligned} \frac{c^{-1}}{2}\big (e(t,x)+{\widetilde{e}}(t,x)\big )\le |\langle \xi \rangle ^{1+\delta }v_{\xi }|^2+|\langle \eta \rangle ^{1+\delta }v_{\eta }|^2\le \frac{c}{2}\big (e(t,x)+{\widetilde{e}}(t,x)\big ). \end{aligned}$$

(2.62)

Note that $v(t,0)=0$ implies

$$\begin{aligned} v_{\xi }(t,0)&=v_{t}(t,0)+v_{x}(t,0)=v_{x}(t,0),~~v_{\eta }(t,0)=v_{t}(t,0)-v_{x}(t,0)\nonumber \\&=-v_{x}(t,0). \end{aligned}$$

(2.63)

In view of (2.56), by (2.63) we have

$$\begin{aligned} p(t,0)&=\psi (\frac{t}{2})\phi (\frac{t}{2})\big (|v_{\xi }(t,0)|^2-|v_{\eta }(t,0)|^2\big )\nonumber \\&=\psi (\frac{t}{2})\phi (\frac{t}{2})\big (|v_{x}(t,0)|^2-|v_{x}(t,0)|^2\big )=0. \end{aligned}$$

(2.64)

In view of (2.57), by (2.63) we also have

$$\begin{aligned}&{\widetilde{p}}(t,0)\nonumber \\&\quad =\psi (\frac{t}{2})\phi (\frac{t}{2})\big (-v^{{T}}_{\xi }(t,0)A_1v_{\xi }(t,0)+v^{{T}}_{\eta }(t,0)A_1v_{\eta }(t,0)\big )\nonumber \\&\qquad +\psi (\frac{t}{2})\phi (\frac{t}{2})\big (v^{{T}}_{\xi }(t,0)A_2v_{\xi }(t,0)+2v^{{T}}_{\eta }(t,0)A_2v_{\xi }(t,0)+v^{{T}}_{\xi }(t,0)A_2v_{\xi }(t,0)\big )\nonumber \\&\qquad +\psi (\frac{t}{2})\phi (\frac{t}{2})\big (-2v^{{T}}_{\xi }(t,0)A_3v_{\eta }(t,0)-v^{{T}}_{\eta }(t,0)A_3v_{\eta }(t,0)-v^{{T}}_{\eta }(t,0)A_3v_{\eta }(t,0)\big )\nonumber \\&\quad =\psi (\frac{t}{2})\phi (\frac{t}{2})\big (-v^{{T}}_{x}(t,0)A_1v_{x}(t,0)+v^{{T}}_{x}(t,0)A_1v_{x}(t,0)\big )\nonumber \\&\qquad +\psi (\frac{t}{2})\phi (\frac{t}{2})\big (v^{{T}}_{x}(t,0)A_2v_{x}(t,0)-2v^{{T}}_{x}(t,0)A_2v_{x}(t,0)+v^{{T}}_{x}(t,0)A_2v_{x}(t,0)\big )\nonumber \\&\qquad +\psi (\frac{t}{2})\phi (\frac{t}{2})\big (2v^{{T}}_{x}(t,0)A_3v_{x}(t,0)-v^{{T}}_{x}(t,0)A_3v_{x}(t,0)-v^{{T}}_{x}(t,0)A_3v_{x}(t,0)\big )\nonumber \\&\quad =0. \end{aligned}$$

(2.65)

By (2.1) and (2.6), we can obtain

$$\begin{aligned} c^{-1}q(t,x)\le \langle \eta \rangle ^{-(1+\delta )}\langle \xi \rangle ^{2+2\delta }|v_{\xi }|^2+\langle \xi \rangle ^{-(1+\delta )}\langle \eta \rangle ^{2+2\delta }|v_{\eta }|^2\le cq(t,x). \end{aligned}$$

(2.66)

By (2.59) and Lemma 2.1, we have

$$\begin{aligned}&|w(t,x)|\nonumber \\&\quad \le C\langle \eta \rangle ^{-(1+\delta )}\langle \xi \rangle ^{2+2\delta }|v_{\xi }|^2(|u|+|u_{\xi }|+|u_{\eta }|)+C\langle \eta \rangle ^{-(1+\delta )}\langle \xi \rangle ^{2+2\delta }|v_{\xi }||v_{\eta }||u_{\xi }|^2\nonumber \\&\qquad +C\langle \xi \rangle ^{1+2\delta }|v_{\xi }|^2|u_{\eta }|^2+{{C\langle \xi \rangle ^{1+2\delta }|v_{\eta }|^2|u_{\xi }|^2}}\nonumber \\&\qquad + C\langle \xi \rangle ^{-(1+\delta )}\langle \eta \rangle ^{2+2\delta }|v_{\eta }|^2(|u|+|u_{\xi }|+|u_{\eta }|)+C\langle \xi \rangle ^{-(1+\delta )}\langle \eta \rangle ^{2+2\delta }|v_{\xi }||v_{\eta }|u_{\eta }|^2\nonumber \\&\qquad +{{C\langle \eta \rangle ^{1+2\delta }|v_{\xi }|^2|u_{\eta }|^2}}+C\langle \eta \rangle ^{1+2\delta }|v_{\eta }|^2|u_{\xi }|^2\nonumber \\&\qquad +C\langle \xi \rangle ^{2+2\delta }|v_{\xi }|^2(|u_{\eta }|+|u_{\xi \eta }|+|u_{\eta \eta }|+|u_{\eta }||u_{\xi }|+|u_{\eta }||Zu_{\xi }|)\nonumber \\&\qquad +C\langle \xi \rangle ^{2+2\delta }|v_{\xi }||v_{\eta }||u_{\xi }|(|u_{\eta }|+|Zu_{\eta }|)+{{C\langle \xi \rangle ^{2+2\delta }|v_{\eta }|^2|u_{\xi }|(|u_{\xi }|+|Zu_{\xi }|)}}\nonumber \\&\qquad +C\langle \eta \rangle ^{2+2\delta }|v_{\eta }|^2(|u_{\xi }|+|u_{\xi \eta }|+|u_{\xi \xi }|+|u_{\xi }||u_{\eta }|+|u_{\xi }||Zu_{\eta }|)\nonumber \\&\qquad +C\langle \eta \rangle ^{2+2\delta }|v_{\xi }||v_{\eta }||u_{\eta }|(|u_{\xi }|+|Zu_{\xi }|)+{{C\langle \eta \rangle ^{2+2\delta }|v_{\xi }|^2|u_{\eta }|(|u_{\eta }|+|Zu_{\eta }|)}}\nonumber \\&\quad \le C|\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }|^2(|u|+|u_{\xi }|+|u_{\eta }|)\nonumber \\&\qquad +C|\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }||\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }||v_{\eta }||u_{\xi }|\nonumber \\&\qquad +C|\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }|^2|\langle \eta \rangle ^{1+\delta }u_{\eta }||u_{\eta }|+C|\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }|^2|\langle \eta \rangle ^{1+\delta }v_{\eta }||v_{\eta }|\nonumber \\&\qquad + C|\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }|^2(|u|+|u_{\xi }|+|u_{\eta }|)\nonumber \\&\qquad +C|\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }||\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }||v_{\xi }||u_{\eta }|\nonumber \\&\qquad +C|\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }|^2|\langle \xi \rangle ^{1+\delta }v_{\xi }||v_{\xi }|+C|\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }|^2||\langle \xi \rangle ^{1+\delta }u_{\xi }||u_{\xi }|\nonumber \\&\qquad +C|\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }|^2\langle \eta \rangle ^{1+\delta }(|u_{\eta }|+|u_{\xi \eta }|+|u_{\eta \eta }|+|u_{\eta }||u_{\xi }|+|u_{\eta }||Zu_{\xi }|)\nonumber \\&\qquad +C|\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }||\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }||\langle \eta \rangle ^{1+\delta }v_{\eta }|(|u_{\eta }|+|Zu_{\eta }|)\nonumber \\&\qquad +C|\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }||\langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }(|u_{\xi }|+|Zu_{\xi }|)||\langle \eta \rangle ^{1+\delta }v_{\eta }||v_{\eta }|\nonumber \\&\qquad +C|\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }|^2\langle \xi \rangle ^{1+\delta }(|u_{\xi }|+|u_{\xi \eta }|+|u_{\xi \xi }|+|u_{\xi }||u_{\eta }|+|u_{\xi }||Zu_{\eta }|)\nonumber \\&\qquad +C|\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }||\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }||\langle \xi \rangle ^{1+\delta }v_{\xi }|(|u_{\xi }|+|Zu_{\xi }|)\nonumber \\&\qquad +C|\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }||\langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }(|u_{\eta }|+|Zu_{\eta }|)||\langle \xi \rangle ^{1+\delta }v_{\xi }||v_{\xi }|. \end{aligned}$$

(2.67)

The combination of (2.53), (2.62), (2.64), (2.65) and (2.66) implies

$$\begin{aligned}&\sup _{0\le s\le t}E(v(s))+ {\mathcal {E}}(v(t))\nonumber \\&\quad \le C E(v(0))+C\int _0^{t}\Vert w(s,\cdot )\Vert _{L^1({\mathbb {R}})}ds\nonumber \\&\qquad +C\int _0^{t}\Vert \langle \xi \rangle ^{2+2\delta }v^{T}_{\xi }G\Vert _{L_{x}^1({\mathbb {R}})}ds+C\int _0^{t}\Vert \langle \eta \rangle ^{2+2\delta }v^{T}_{\eta } G\Vert _{L_{x}^1({\mathbb {R}})}ds. \end{aligned}$$

(2.68)

It follows from (2.67), Hölder inequality and Lemma 2.3 that

$$\begin{aligned}&\int _0^{t}\Vert w(s,\cdot )\Vert _{L^1({\mathbb {R}})}ds\nonumber \\&\quad \le C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }\Vert _{L^2_{s,x}}^2(\Vert u\Vert _{L^{\infty }_{s,x}}+\Vert u_{\xi }\Vert _{L^{\infty }_{s,x}}+\Vert u_{\eta }\Vert _{L^{\infty }_{s,x}})\nonumber \\&\qquad +C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }\Vert _{L^{2}_{s,x}}\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L^{2}_{s}L^{\infty }_{x}}\Vert v_{\eta }\Vert _{L^{\infty }_{s}L^{2}_{x}}\Vert u_{\xi }\Vert _{L^{\infty }_{s,x}}\nonumber \\&\qquad +C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }\Vert _{L^{2}_{s,x}}^2\Vert \langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L^{\infty }_{s,x}}\Vert u_{\eta }\Vert _{L^{\infty }_{s,x}}\nonumber \\&\qquad +C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L^{2}_{s}L^{\infty }_{x}}^2\Vert \langle \eta \rangle ^{1+\delta }v_{\eta }\Vert _{L^{\infty }_{s}L^{2}_{x}}\Vert v_{\eta }\Vert _{L^{\infty }_{s}L^{2}_{x}}\nonumber \\&\qquad + C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }\Vert _{L^2_{s,x}}^2(\Vert u\Vert _{L^{\infty }_{s,x}}+\Vert u_{\xi }\Vert _{L^{\infty }_{s,x}}+\Vert u_{\eta }\Vert _{L^{\infty }_{s,x}})\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }\Vert _{L^{2}_{s,x}}\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L^{2}_{s}L^{\infty }_{x}}\Vert v_{\xi }\Vert _{L^{\infty }_{s}L^{2}_{x}}\Vert u_{\eta }\Vert _{L^{\infty }_{s,x}}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L^{2}_{s}L^{\infty }_{x}}^2\Vert \langle \xi \rangle ^{1+\delta }v_{\xi }\Vert _{L^{\infty }_{s}L^{2}_{x}}\Vert v_{\xi }\Vert _{L^{\infty }_{s}L^{2}_{x}}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }\Vert _{L^2_{s,x}}^2\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L^{\infty }_{s,x}}\Vert u_{\xi }\Vert _{L^{\infty }_{s,x}}\nonumber \\&\qquad +C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }\Vert _{L^2_{s,x}}^2\Vert \langle \eta \rangle ^{1+\delta }(|u_{\eta }|+|u_{\xi \eta }|+|u_{\eta \eta }|+|u_{\eta }||u_{\xi }|\nonumber \\&\qquad +|u_{\eta }||Zu_{\xi }|)\Vert _{L^{\infty }_{s,x}}\nonumber \\&\qquad +C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }v_{\xi }\Vert _{L^2_{s,x}}\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L^2_{s}L^{\infty }_{x}}\Vert \langle \eta \rangle ^{1+\delta }v_{\eta }\Vert _{L^{\infty }_{s}L^2_{x}}\Vert (|u_{\eta }|\nonumber \\&\qquad +|Zu_{\eta }|\Vert _{L^{\infty }_{s,x}}\nonumber \\&\qquad +C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L^{2}_{s}L^{\infty }_{x}}\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }(|u_{\xi }|\nonumber \\&\qquad +|Zu_{\xi }|)\Vert _{L^{2}_{s}L^{\infty }_{x}}\Vert \langle \eta \rangle ^{1+\delta }v_{\eta }\Vert _{L^{\infty }_{s}L^{2}_{x}}\Vert v_{\eta }\Vert _{L^{\infty }_{s}L^{2}_{x}}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }\Vert _{_{L^2_{s,x}}}^2\Vert \langle \xi \rangle ^{1+\delta }(|u_{\xi }|+|u_{\xi \eta }|+|u_{\xi \xi }|+|u_{\xi }||u_{\eta }|\nonumber \\&\qquad +|u_{\xi }||Zu_{\eta }|)\Vert _{_{L^{\infty }_{s,x}}}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }v_{\eta }\Vert _{L^2_{s,x}}\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L^{2}_{s}L^{\infty }_{x}}\Vert \langle \xi \rangle ^{1+\delta }v_{\xi }\Vert _{L^{\infty }_{s}L^{2}_{x}}\Vert (|u_{\xi }|\nonumber \\&\qquad +|Zu_{\xi }|)\Vert _{L^{\infty }_{s,x}}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L^{2}_{s}L^{\infty }_{x}}\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }(|u_{\eta }|\nonumber \\&\qquad +|Zu_{\eta }|)\Vert _{L^{2}_{s}L^{\infty }_{x}}\Vert \langle \xi \rangle ^{1+\delta }v_{\xi }\Vert _{L^{\infty }_{s}L^{2}_{x}}\Vert v_{\xi }\Vert _{L^{\infty }_{s}L^{2}_{x}}\nonumber \\&\quad \le C\sup _{0\le s\le t}\big (E_3^{1/2}(u(s))+E_3(u(s))\big ){\mathcal {E}}(v(t))+C{\mathcal {E}}_3(u(t))\sup _{0\le s\le t} {E}(v(s)). \end{aligned}$$

(2.69)

Finally, by (2.49), (2.68) and (2.69), we have

$$\begin{aligned}&\sup _{0\le s\le t}E(v(s))+ {\mathcal {E}}(v(t))\nonumber \\&\quad \le C E(v(0))+C(\varepsilon _1+\varepsilon ^2_1)\big (\sup _{0\le s\le t}E(v(s))+ {\mathcal {E}}(v(t))\big )\nonumber \\&\qquad +C\int _0^{t}\Vert \langle \xi \rangle ^{2+2\delta }v^{T}_{\xi } G\Vert _{L_{x}^1({\mathbb {R}})}ds+C\int _0^{t}\Vert \langle \eta \rangle ^{2+2\delta }v^{T}_{\eta } G\Vert _{L_{x}^1({\mathbb {R}})}ds. \end{aligned}$$

(2.70)

If $\varepsilon _1$ is sufficiently small, we can obtain (2.50). $\square$

Remark 2.2

In the above proof, we note that in the deriving of equalities (2.64) and (2.65) concerning boundary terms, the homogeneous boundary condition $v(t,0)=0$ plays an elementary and key role. Lemma 2.5 will be directly used in the quasilinear problem, once we use the time derivative, which preserves the homogeneous boundary conditions, as the commuting vector field.

The following lemma will be used in the proof of scattering part of Theorem 1.1. The proof can be found in Lemma 2.2 of [19].

Lemma 2.6

If $H\in L^1({\mathbb {R}}^{+};L^2({\mathbb {R}}^{+}))$, i.e.,

$$\begin{aligned} \int _0^{+\infty }\Vert H(t,\cdot )\Vert _{L^2({\mathbb {R}}^{+})}dt<+\infty . \end{aligned}$$

(2.71)

Then the global solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{\xi \eta }(t,x)=H(t,x), t>0,x>0,\\ u(t,0)=0, t\ge 0,\\ t=0: u=u_0(x), u_{t}=u_{1}(x), x\ge 0, \end{array}\right. } \end{aligned}$$

(2.72)

where $(u_0,u_1)\in {\dot{H}}^1({\mathbb {R}}^{+})\times L^2({\mathbb {R}}^{+})$, is asymptotically free in the energy sense.

The following lemma is the key part in the proof of rigidity part of Theorem 1.1.

Lemma 2.7

Assume that $v: {\mathbb {R}}^{+}\times {\mathbb {R}}^{+}\longrightarrow {\mathbb {R}}^n$ satisfies the following system of linear wave equations

$$\begin{aligned} v_{\xi \eta }&=A_1(u,u_{\xi },u_{\eta })v_{\xi \eta } +A_2(u,u_{\xi },u_{\eta })v_{\xi \xi }+A_3(u,u_{\xi },u_{\eta })v_{\eta \eta }+G, \end{aligned}$$

(2.73)

and homogeneous boundary condition $v(t,0)=0$. Here $A_1,A_2$, $A_3$ are symmetric and satisfies (1.4), (1.5′), (1.6′), respectively, u and $G: {\mathbb {R}}^{+}\times {\mathbb {R}}^{+}\longrightarrow {\mathbb {R}}^n$ are some given vector valued functions of (t, x), and

$$\begin{aligned} \varepsilon _1=\sup _{0\le s\le t}E_3(u(s)) \end{aligned}$$

(2.74)

is sufficiently small. Then it holds that

$$\begin{aligned}&\sup _{0\le s\le t}{E}^{c}(v(s))+ {{\mathcal {E}}}^{c}(v(t))\nonumber \\&\quad \le C E^{c}(v(t))+C\int _0^{t}\Vert v^{T}_{\xi }G\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds+C\int _0^{t}\Vert v^{T}_{\eta } G\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds. \end{aligned}$$

(2.75)

Proof

Multiply $2\psi (-\eta )v^{{T}}_{\xi }$ on both sides of (2.73). Noting the symmetry of $A_1, A_2$ and $A_3$, by Leibniz’s rule we obtain

$$\begin{aligned}&\big (\psi (-\eta )|v_{\xi }|^2\big )_{\eta }+\psi '(-\eta )|v_{\xi }|^2\nonumber \\&\quad =\big (\psi (-\eta )v_{\xi }^{{T}}A_1v_{\xi }\big )_{\eta }+\psi '(-\eta )v_{\xi }^{{T}}A_1v_{\xi } -\psi (-\eta )v_{\xi }^{{T}}\partial _{\eta }A_1v_{\xi }\nonumber \\&\qquad +\big (\psi (-\eta )v_{\xi }^{{T}}A_2v_{\xi }\big )_{\xi } -\psi (-\eta )v_{\xi }^{{T}}\partial _{\xi }A_2v_{\xi }\nonumber \\&\qquad +\big (2\psi (-\eta )v_{\xi }^{{T}}A_3v_{\eta }\big )_{\eta }+2\psi '(-\eta )v_{\xi }^{{T}}A_3v_{\eta } -2\psi (-\eta )v_{\xi }^{{T}}\partial _{\eta }A_3v_{\eta }\nonumber \\&\qquad -\big (\psi (-\eta )v_{\eta }^{{T}}A_3v_{\eta }\big )_{\xi } +\psi (-\eta )v_{\eta }^{{T}}\partial _{\xi }A_3v_{\eta }+2\psi (-\eta )v^{{T}}_{\xi }G. \end{aligned}$$

(2.76)

Similarly, multiply $2\psi (-\xi )v^{{T}}_{\eta }$ on both sides of (2.73). The symmetry of $A_1, A_2$ and $A_3$ and Leibniz’s rule also imply

$$\begin{aligned}&\big (\psi (-\xi )|v_{\eta }|^2\big )_{\xi }+\psi '(-\xi )|v_{\eta }|^2\nonumber \\&\quad =\big (\psi (-\xi )v_{\eta }^{{T}}A_1v_{\eta }\big )_{\xi }+\psi '(-\xi )v_{\eta }^{{T}}A_1v_{\eta } -\psi (-\xi )v_{\eta }^{{T}}\partial _{\xi }A_1v_{\eta }\nonumber \\&\qquad +\big (2\psi (-\xi )v_{\eta }^{{T}}A_2v_{\xi }\big )_{\xi }+2\psi '(-\xi )v_{\eta }^{{T}}A_2v_{\xi } -2\psi (-\xi )v_{\eta }^{{T}}\partial _{\xi }A_2v_{\xi }\nonumber \\&\qquad -\big (\psi (-\xi )v_{\xi }^{{T}}A_2v_{\xi }\big )_{\eta } +\psi (-\xi )v_{\xi }^{{T}}\partial _{\eta }A_2v_{\xi }\nonumber \\&\qquad +\big (\psi (-\xi )v_{\eta }^{{T}}A_3v_{\eta }\big )_{\eta } -\psi (-\xi )v_{\eta }^{{T}}\partial _{\eta }A_3v_{\eta }+2\psi (-\xi )v^{{T}}_{\eta }G. \end{aligned}$$

(2.77)

Fix $0\le \tau <t$. Integrating on $[\tau ,t]\times {\mathbb {R}}^{+}$ on both sides of (2.76) and (2.77), by the fundamental theorem of calculus we get

$$\begin{aligned}&\int _{{\mathbb {R}}^{+}}\big (e(\tau ,x)+{\widetilde{e}}(\tau ,x)\big )dx-\int _{\tau }^{t}\big (p(s,0)+{\widetilde{p}}(s,0)\big )ds-\int _{\tau }^{t}\!\!\int _{{\mathbb {R}}^{+}}q(s,x)dxds\nonumber \\&\quad =\int _{{\mathbb {R}}^{+}}\big (e(t,x)+{\widetilde{e}}(t,x)\big )dx-\int _{\tau }^{t}\!\!\int _{{\mathbb {R}}^{+}}w(s,x)dxds\nonumber \\&\qquad -2\int _0^{t}\!\!\int _{{\mathbb {R}}^{+}}\psi (-\eta )v^{{T}}_{\xi }Gdxds -2\int _0^{t}\!\!\int _{{\mathbb {R}}^{+}}\psi (-\xi )v^{{T}}_{\eta }Gdxds, \end{aligned}$$

(2.78)

where

$$\begin{aligned} e&=\psi (-\eta )|v_{\xi }|^2+\psi (-\xi )|v_{\eta }|^2, \end{aligned}$$

(2.79)

$$\begin{aligned} {\widetilde{e}}&=-\psi (-\eta )v_{\xi }^{{T}}A_1v_{\xi }-\psi (-\eta )v_{\xi }^{{T}}A_2v_{\xi } -2\psi (-\eta )v_{\xi }^{{T}}A_3v_{\eta }+{{\psi (-\eta )v_{\eta }^{{T}}A_3v_{\eta }}}\nonumber \\&\quad -\psi (-\xi )v_{\eta }^{{T}}A_1v_{\eta } -2\psi (-\xi )v_{\eta }^{{T}}A_2v_{\xi } +{{\psi (-\xi )v_{\xi }^{{T}}A_2v_{\xi }}} -\psi (-\xi )v_{\eta }^{{T}}A_3v_{\eta }, \end{aligned}$$

(2.80)

$$\begin{aligned} p&=\psi (-\eta )|v_{\xi }|^2-\psi (-\xi )|v_{\eta }|^2, \end{aligned}$$

(2.81)

$$\begin{aligned} {\widetilde{p}}&=-\psi (-\eta )v_{\xi }^{{T}}A_1v_{\xi }+\psi (-\eta )v_{\xi }^{{T}}A_2v_{\xi } -2\psi (-\eta )v_{\xi }^{{T}}A_3v_{\eta }-\psi (-\eta )v_{\eta }^{{T}}A_3v_{\eta }\nonumber \\&\quad +\psi (-\xi )v_{\eta }^{{T}}A_1v_{\eta } +2\psi (-\xi )v_{\eta }^{{T}}A_2v_{\xi } +\psi (-\xi )v_{\xi }^{{T}}A_2v_{\xi } -\psi (-\xi )v_{\eta }^{{T}}A_3v_{\eta }, \end{aligned}$$

(2.82)

$$\begin{aligned} q&=\psi '(-\eta )|v_{\xi }|^2+\psi '(-\xi )|v_{\eta }|^2, \end{aligned}$$

(2.83)

and

$$\begin{aligned} w&=\psi '(-\eta )v_{\xi }^{{T}}A_1v_{\xi }+2\psi '(-\eta )v_{\xi }^{{T}}A_3v_{\eta } +\psi '(-\xi )v_{\eta }^{{T}}A_1v_{\eta }+2\psi '(-\xi )v_{\eta }^{{T}}A_2v_{\xi }\nonumber \\&\quad -\psi (-\eta )v_{\xi }^{{T}}\partial _{\eta }A_1v_{\xi }-\psi (-\eta )v_{\xi }^{{T}}\partial _{\xi }A_2v_{\xi } -2\psi (-\eta )v_{\xi }^{{T}}\partial _{\eta }A_3v_{\eta }\nonumber \\&\quad +{{\psi (-\eta )v_{\eta }^{{T}}\partial _{\xi }A_3v_{\eta }}} \nonumber \\&\quad -\psi (-\xi )v_{\eta }^{{T}}\partial _{\xi }A_1v_{\eta } -2\psi (-\xi )v_{\eta }^{{T}}\partial _{\xi }A_2v_{\xi } +{{\psi (-\xi )\phi (\eta )v_{\xi }^{{T}}\partial _{\eta }A_2v_{\xi }}}\nonumber \\&\quad -\psi (-\xi )v_{\eta }^{{T}}\partial _{\eta }A_3v_{\eta }. \end{aligned}$$

(2.84)

By (2.5), we have

$$\begin{aligned} c^{-1}e(t,x)\le |v_{\xi }|^2+|v_{\eta }|^2\le ce(t,x) \end{aligned}$$

(2.85)

It follows from (2.5), (1.4), (1.5′), (1.6′) and Lemma 2.3 that

$$\begin{aligned} |{\widetilde{e}}(t,x)|&\le C|v_{\xi }^{{T}}A_1v_{\xi }|+C|v_{\xi }^{{T}}A_2v_{\xi }| +C|v_{\xi }^{{T}}A_3v_{\eta }|+C|v_{\eta }^{{T}}A_3v_{\eta }|\nonumber \\&\quad +C|v_{\eta }^{{T}}A_1v_{\eta }| +C|v_{\eta }^{{T}}A_2v_{\xi }| +C|v_{\xi }^{{T}}A_2v_{\xi }| +C|v_{\eta }^{{T}}A_3v_{\eta }|\nonumber \\&\le C\big (E_3^{1/2}(u(t))+E_3(u(t))\big ) \big (|v_{\xi }|^2+ |v_{\eta }|^2\big ). \end{aligned}$$

(2.86)

Noting (2.74), if $\varepsilon _1$ is sufficiently small, we can get

$$\begin{aligned} \frac{c^{-1}}{2}\big (e(t,x)+{\widetilde{e}}(t,x)\big )\le |v_{\xi }|^2+|v_{\eta }|^2\le \frac{c}{2}\big (e(t,x)+{\widetilde{e}}(t,x)\big ). \end{aligned}$$

(2.87)

Note that $v(t,0)=0$ implies

$$\begin{aligned} v_{\xi }(t,0)=v_{t}(t,0)+v_{x}(t,0)=v_{x}(t,0),~~v_{\eta }(t,0)=v_{t}(t,0)-v_{x}(t,0)=-v_{x}(t,0). \end{aligned}$$

(2.88)

In view of (2.81), by (2.88) we have

$$\begin{aligned} p(t,0)=\psi (-\frac{t}{2})\big (|v_{\xi }(t,0)|^2-|v_{\eta }(t,0)|^2\big )=\psi (-\frac{t}{2})\big (|v_{x}(t,0)|^2-|v_{x}(t,0)|^2\big )=0. \end{aligned}$$

(2.89)

In view of (2.82), by (2.88) we also have

$$\begin{aligned}&{\widetilde{p}}(t,0)\nonumber \\&\quad =\psi (-\frac{t}{2})\big (-v^{{T}}_{\xi }(t,0)A_1v_{\xi }(t,0)+v^{{T}}_{\eta }(t,0)A_1v_{\eta }(t,0)\big )\nonumber \\&\qquad +\psi (-\frac{t}{2})\big (v^{{T}}_{\xi }(t,0)A_2v_{\xi }(t,0)+2v^{{T}}_{\eta }(t,0)A_2v_{\xi }(t,0)+v^{{T}}_{\xi }(t,0)A_2v_{\xi }(t,0)\big )\nonumber \\&\qquad +\psi (-\frac{t}{2})\big (-2v^{{T}}_{\xi }(t,0)A_3v_{\eta }(t,0)-v^{{T}}_{\eta }(t,0)A_3v_{\eta }(t,0)-v^{{T}}_{\eta }(t,0)A_3v_{\eta }(t,0)\big )\nonumber \\&\quad =\psi (-\frac{t}{2})\big (-v^{{T}}_{x}(t,0)A_1v_{x}(t,0)+v^{{T}}_{x}(t,0)A_1v_{x}(t,0)\big )\nonumber \\&\qquad +\psi (-\frac{t}{2})\big (v^{{T}}_{x}(t,0)A_2v_{x}(t,0)-2v^{{T}}_{x}(t,0)A_2v_{x}(t,0)+v^{{T}}_{x}(t,0)A_2v_{x}(t,0)\big )\nonumber \\&\qquad +\psi (-\frac{t}{2})\big (2v^{{T}}_{x}(t,0)A_3v_{x}(t,0)-v^{{T}}_{x}(t,0)A_3v_{x}(t,0)-v^{{T}}_{x}(t,0)A_3v_{x}(t,0)\big )\nonumber \\&\quad =0. \end{aligned}$$

(2.90)

By (2.6), we can obtain

$$\begin{aligned} {{-}}c^{-1}q(t,x)\le \langle \eta \rangle ^{-(1+\delta )}|v_{\xi }|^2+\langle \xi \rangle ^{-(1+\delta )}|v_{\eta }|^2\le {{-}}cq(t,x). \end{aligned}$$

(2.91)

By (2.84), Lemmas 2.1 and 2.3, we have

$$\begin{aligned} |w(t,x)|&\le C\langle \eta \rangle ^{-(1+\delta )}|v_{\xi }|^2(|u|+|u_{\xi }|+|u_{\eta }|)+C\langle \eta \rangle ^{-(1+\delta )}|v_{\xi }||v_{\eta }||u_{\xi }|^2\nonumber \\&\quad + C\langle \xi \rangle ^{-(1+\delta )}|v_{\eta }|^2(|u|+|u_{\xi }|+|u_{\eta }|)+C\langle \xi \rangle ^{-(1+\delta )}|v_{\xi }||v_{\eta }|u_{\eta }|^2\nonumber \\&\quad +C|v_{\xi }|^2(|u_{\eta }|+|u_{\xi \eta }|+|u_{\eta \eta }|+|u_{\eta }||u_{\xi }|+|u_{\eta }||Zu_{\xi }|)\nonumber \\&\quad +C|v_{\xi }||v_{\eta }||u_{\xi }|(|u_{\eta }|+|Zu_{\eta }|)+{{C|v_{\eta }|^2|u_{\xi }|(|u_{\xi }|+|Zu_{\xi }|)}}\nonumber \\&\quad +C|v_{\eta }|^2(|u_{\xi }|+|u_{\xi \eta }|+|u_{\xi \xi }|+|u_{\xi }||u_{\eta }|+|u_{\xi }||Zu_{\eta }|)\nonumber \\&\quad +C|v_{\xi }||v_{\eta }||u_{\eta }|(|u_{\xi }|+|Zu_{\xi }|)+{{C|v_{\xi }|^2|u_{\eta }|(|u_{\eta }|+|Zu_{\eta }|)}}\nonumber \\&\le C(|\langle \eta \rangle ^{-\frac{1+\delta }{2}}v_{\xi }|^2+|\langle \xi \rangle ^{-\frac{1+\delta }{2}}v_{\eta }|^2)\big (E_3^{1/2}(u(t))+E_3(u(t))\big ). \end{aligned}$$

(2.92)

The combination of (2.78), (2.87), (2.89), (2.90), (2.91) and (2.92) implies

$$\begin{aligned}&E^c(v(\tau ))+ \int _{\tau }^{t}\big (\langle \eta \rangle ^{-(1+\delta )}|v_{\xi }|^2+\langle \xi \rangle ^{-(1+\delta )}|v_{\eta }|^2\big )ds\nonumber \\&\quad \le C E^{c}(v(t))+C\int _0^{t}\Vert w(s,\cdot )\Vert _{L^1({\mathbb {R}})}ds+C\int _0^{t}\Vert v^{T}_{\xi }G\Vert _{L_{x}^1({\mathbb {R}})}ds\nonumber \\&\qquad +C\int _0^{t}\Vert v^{T}_{\eta } G\Vert _{L_{x}^1({\mathbb {R}})}ds\nonumber \\&\quad \le C E^{c}(v(t))+C(\varepsilon _1+\varepsilon _1^2){\mathcal {E}}^c(v(t))+C\int _0^{t}\Vert v^{T}_{\xi }G\Vert _{L_{x}^1({\mathbb {R}})}ds\nonumber \\&\qquad +C\int _0^{t}\Vert v^{T}_{\eta } G\Vert _{L_{x}^1({\mathbb {R}})}ds \end{aligned}$$

(2.93)

for any $0\le \tau <t$. By the arbitrariness of $\tau$, we get

$$\begin{aligned}&\sup _{0\le s\le t}E^c(v(s))+ {\mathcal {E}}^c(v(t))\nonumber \\&\quad \le C E^{c}(v(t))+C(\varepsilon _1+\varepsilon _1^2){\mathcal {E}}^c(v(t))\nonumber \\&\qquad +C\int _0^{t}\Vert v^{T}_{\xi }G\Vert _{L_{x}^1({\mathbb {R}})}ds+C\int _0^{t}\Vert v^{T}_{\eta } G\Vert _{L_{x}^1({\mathbb {R}})}ds. \end{aligned}$$

(2.94)

If $\varepsilon _1$ is sufficiently small, we can obtain (2.75). $\square$

3 Proof of Theorem 1.1

Now we will prove Theorem 1.1 by some bootstrap argument. Assume that u is a classical solution to the initial-boundary value problem (1.3), (1.8), (1.9). We will first show that there exist positive constants $\varepsilon _0$ and A such that

$$\begin{aligned} \sup _{0\le s\le t}E_3(u(s))+{\mathcal {E}}_3(u(t))\le A^2\varepsilon ^2 \end{aligned}$$

(3.1)

under the assumption

$$\begin{aligned} \sup _{0\le s\le t}E_3(u(s))+{\mathcal {E}}_3(u(t))\le 4A^2\varepsilon ^2, \end{aligned}$$

(3.2)

where $0<\varepsilon \le \varepsilon _0$. Then we will give the proof of asymptotic behavior part (scattering and rigidity) of Theorem 1.1, based on (3.1) and other related estimates.

3.1 Energy Estimates

In view of Lemma 2.4, we have

$$\begin{aligned} \sup _{0\le s\le t}E_3(u(s))\le C\sup _{0\le s\le t}{E}_3^2(u(s))+C\sup _{0\le s\le t}{\widetilde{E}}_3(u(s)) \end{aligned}$$

(3.3)

and

$$\begin{aligned} {\mathcal {E}}_3(u(t))\le C\sup _{0\le s\le t}{E}_3(u(s)){{\mathcal {E}}}_3(u(t))+C\widetilde{{\mathcal {E}}}_3(u(t)). \end{aligned}$$

(3.4)

Thus in order to estimate $\sup \limits _{0\le s\le t}{E}_3(u(s))$ and ${{\mathcal {E}}}_3(u(t))$, our task is to control $\sup \limits _{0\le s\le t}{\widetilde{E}}_3(u(s))$ and $\widetilde{{\mathcal {E}}}_3(u(t))$.

In view of the system (1.3), we have

$$\begin{aligned} u_{t\xi \eta }&=A_1u_{t\xi \eta }+A_2u_{t\xi \xi }+A_3u_{t\eta \eta }+G_2, \end{aligned}$$

(3.5)

$$\begin{aligned} u_{tt\xi \eta }&=A_1u_{tt\xi \eta }+A_2u_{tt\xi \xi }+A_3u_{tt\eta \eta }+G_3, \end{aligned}$$

(3.6)

where

$$\begin{aligned} G_2&=\partial _tA_1u_{\xi \eta }+\partial _tA_2u_{\xi \xi } +\partial _tA_3u_{\eta \eta }+\partial _tF,\nonumber \\ G_3&=2\partial _tA_1u_{t\xi \eta }+2\partial _tA_2u_{t\xi \xi } +2\partial _tA_3u_{t\eta \eta }+\partial ^2_tA_1u_{\xi \eta }\nonumber \\&\quad +\partial ^2_tA_2u_{\xi \xi }+\partial ^2_tA_3u_{\eta \eta }+\partial ^2_tF. \end{aligned}$$

(3.7)

In view of the homogeneous boundary condition (1.8), we have $u_{t}(t,0)=0, u_{tt}(t,0)=0$. Then it follows from (3.5), (3.6) and Lemma 2.5 that

$$\begin{aligned}&\sup _{0\le s\le t}\widetilde{{E}}_3(u(s))+ {\widetilde{{\mathcal {E}}}_3}(u(t))\nonumber \\&\quad \le C E_3(u(0))+C\int _0^{t}\Vert \langle \xi \rangle ^{2+2\delta }u^{T}_{t\xi }G_2\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds\nonumber \\&\qquad +C\int _0^{t}\Vert \langle \eta \rangle ^{2+2\delta }u^{T}_{t\eta } G_2\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds\nonumber \\&\qquad +C\int _0^{t}\Vert \langle \xi \rangle ^{2+2\delta }u^{T}_{tt\xi }G_3\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds+C\int _0^{t}\Vert \langle \eta \rangle ^{2+2\delta }u^{T}_{tt\eta } G_3\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds. \end{aligned}$$

(3.8)

Lemma 2.2 implies

$$\begin{aligned} |G_2|+|G_3|\le C\sum _{|b|\le 1}|Z^{b}u_{\xi }| \sum _{|c|\le 2}|Z^{c}u_{\eta }|+C\sum _{|b|\le 2}|Z^{b}u_{\xi }| \sum _{|c|\le 1}|Z^{c}u_{\eta }|. \end{aligned}$$

(3.9)

Then by (3.9) and Lemma 2.3 we obtain

$$\begin{aligned}&\int _0^{t}\Vert \langle \xi \rangle ^{2+2\delta }u^{T}_{tt\xi }G_3\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds\nonumber \\&\quad \le C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{tt\xi }\Vert _{L^2_{s,x}} \sum _{|b|\le 1}\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }Z^{b}u_{\xi }\Vert _{L^2_{s}L^{\infty }_{x}} \nonumber \\&\qquad \sum _{|c|\le 2}\Vert \langle \eta \rangle ^{1+\delta }Z^{c}u_{\eta }\Vert _{L^{\infty }_{s}L^2_{x}}\nonumber \\&\qquad +C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }u_{tt\xi }\Vert _{L^2_{s,x}} \sum _{|b|\le 2}\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}}\langle \xi \rangle ^{1+\delta }Z^{b}u_{\xi }\Vert _{L^2_{s,x}}\nonumber \\&\qquad \sum _{|c|\le 1}\Vert \langle \eta \rangle ^{1+\delta }Z^{c}u_{\eta }\Vert _{L^{\infty }_{s,x}}\nonumber \\&\quad \le C\sup _{0\le s\le t}E_3^{1/2}(u(s)){\mathcal {E}}_3(u(t)) \end{aligned}$$

(3.10)

and

$$\begin{aligned}&\int _0^{t}\Vert \langle \eta \rangle ^{2+2\delta }u^{T}_{tt\eta }G_3\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds\nonumber \\&\quad \le C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{tt\eta }\Vert _{L^2_{s,x}} \sum _{|c|\le 1}\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }Z^{b}u_{\eta }\Vert _{L^2_{s}L^{\infty }_{x}} \nonumber \\&\qquad \sum _{|b|\le 2}\Vert \langle \xi \rangle ^{1+\delta }Z^{c}u_{\xi }\Vert _{L^{\infty }_{s}L^2_{x}}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }u_{tt\eta }\Vert _{L^2_{s,x}} \sum _{|c|\le 2}\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}}\langle \eta \rangle ^{1+\delta }Z^{b}u_{\eta }\Vert _{L^2_{s,x}}\nonumber \\&\qquad \sum _{|b|\le 1}\Vert \langle \xi \rangle ^{1+\delta }Z^{c}u_{\xi }\Vert _{L^{\infty }_{s,x}}\nonumber \\&\quad \le C\sup _{0\le s\le t}E_3^{1/2}(u(s)){\mathcal {E}}_3(u(t)). \end{aligned}$$

(3.11)

Similarly, we also have

$$\begin{aligned}&\int _0^{t}\Vert \langle \xi \rangle ^{2+2\delta }u^{T}_{t\xi }G_2\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds+\int _0^{t}\Vert \langle \eta \rangle ^{2+2\delta }u^{T}_{t\eta }G_2\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds\nonumber \\&\quad \le C\sup _{0\le s\le t}E_3^{1/2}(u(s)){\mathcal {E}}_3(u(t)) \end{aligned}$$

(3.12)

Thus the combination of (3.8), (3.10), (3.11) and (3.12) gives

$$\begin{aligned} \sup _{0\le s\le t}\widetilde{{E}}_3(u(s))+ {\widetilde{{\mathcal {E}}}_3}(u(t))\le C E_3(u(0))+C\sup _{0\le s\le t}E_3^{1/2}(u(s)){\mathcal {E}}_3(u(t)). \end{aligned}$$

(3.13)

Finally, in view of (3.3), (3.4) and (3.13), we have

$$\begin{aligned}&\sup _{0\le s\le t}{{E}}_3(u(s))+ {{{\mathcal {E}}}}_3(u(t))\nonumber \\&\quad \le C E_3(u(0))+C\big (\sup _{0\le s\le t}E_3^{1/2}(u(s))+\sup _{0\le s\le t}E_3(u(s))\big ){\mathcal {E}}_3(u(t))\nonumber \\&\qquad +C\sup _{0\le s\le t}E_3^{2}(u(s)). \end{aligned}$$

(3.14)

3.2 Global Existence

Under the assumption (3.2), by (3.14) we have

$$\begin{aligned} \sup _{0\le s\le t}E_3(u(s))+{\mathcal {E}}_3(u(t))\le C_1\varepsilon ^2+8C_1A^3\varepsilon ^3+16C_1A^4\varepsilon ^4. \end{aligned}$$

(3.15)

Assume that

$$\begin{aligned} E_3(u(0))\le {C_2}\varepsilon ^2. \end{aligned}$$

(3.16)

Taking $A^2=4\max \{C_1,{C_2}\}$ and $\varepsilon _0$ so small that

$$\begin{aligned} 32C_1A\varepsilon _0+64C_1A^2\varepsilon ^2_0\le 1, \end{aligned}$$

(3.17)

for any $\varepsilon$ with $0<\varepsilon \le \varepsilon _0$, we have

$$\begin{aligned} \sup _{0\le s\le t}E_3(u(s))+{\mathcal {E}}_3(u(t))\le A^2\varepsilon ^2, \end{aligned}$$

(3.18)

which completes the proof of global existence part of Theorem 1.1.

3.3 Scattering

Assume that u is the global solution to the initial-boundary value problem (1.3), (1.8), (1.9). Let

$$\begin{aligned} H=(I-A_1)^{-1}\big (A_2u_{\xi \xi }+A_3u_{\eta \eta }+F\big ). \end{aligned}$$

(3.19)

Then we have

$$\begin{aligned} u_{\xi \eta }=H. \end{aligned}$$

(3.20)

According to Lemma 2.6, in order to show that u will scatter, it is sufficient to verity

$$\begin{aligned} \int _0^{+\infty }\Vert H(t,\cdot )\Vert _{L^2({\mathbb {R}}^{+})}dt<+\infty . \end{aligned}$$

(3.21)

We first have

$$\begin{aligned} (I-A_1)H=A_2u_{\xi \xi }+A_3u_{\eta \eta }+F. \end{aligned}$$

(3.22)

Thus it follows from (3.22), (1.5′), (1.6′) and (1.7) that

$$\begin{aligned}&|(I-A_1)H|\nonumber \\&\quad \le C|u_{\eta }||u_{\xi \xi }|+C|u_{\xi }||u_{\eta \eta }|+C|u_{\xi }||u_{\eta }|\nonumber \\&\quad \le C|\langle \xi \rangle ^{-1-\delta }\langle \eta \rangle ^{-1-\delta }\big (|\langle \eta \rangle ^{1+\delta }u_{\eta }||\langle \xi \rangle ^{1+\delta }u_{\xi \xi }|+C|\langle \xi \rangle ^{1+\delta }u_{\xi }||\langle \eta \rangle ^{1+\delta }u_{\eta \eta }|\nonumber \\&\qquad +C|\langle \xi \rangle ^{1+\delta }u_{\xi }||\langle \eta \rangle ^{1+\delta }u_{\eta }|\big )\nonumber \\&\quad \le C\langle t\rangle ^{-1-\delta }\big (|\langle \eta \rangle ^{1+\delta }u_{\eta }||\langle \xi \rangle ^{1+\delta }u_{\xi \xi }|+C|\langle \xi \rangle ^{1+\delta }u_{\xi }||\langle \eta \rangle ^{1+\delta }u_{\eta \eta }|\nonumber \\&\qquad +C|\langle \xi \rangle ^{1+\delta }u_{\xi }||\langle \eta \rangle ^{1+\delta }u_{\eta }|\big ). \end{aligned}$$

(3.23)

On the other hand, by (1.4), Lemma 2.3 and (3.18), we have

$$\begin{aligned} |A_1H|\le C(|u|+|u_{\xi }|+|u_{\eta }|)|H|\le CE_3^{1/2}(u(t))|H|\le CA\varepsilon _0 |H|\le \frac{1}{2}|H|, \end{aligned}$$

(3.24)

if $\varepsilon _0$ is sufficiently small. Then (3.24) implies

$$\begin{aligned} |(I-A_1)H|\ge |H|-|A_1H|\ge \frac{1}{2}|H|. \end{aligned}$$

(3.25)

Now it follows from (3.23), (3.25), Lemma 2.3 and (3.18) that

$$\begin{aligned}&\Vert H(t,\cdot )\Vert _{L^2({\mathbb {R}}^{+})}\nonumber \\&\quad \le C\langle t\rangle ^{-1-\delta }\big (\Vert \langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \xi \rangle ^{1+\delta }u_{\xi \xi }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\qquad +C\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \eta \rangle ^{1+\delta }u_{\eta \eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\big )\nonumber \\&\qquad +C\langle t\rangle ^{-1-\delta }\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L_{x}^{\infty }({\mathbb {R}}^{+})}\Vert \langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L_{x}^2({\mathbb {R}}^{+})}\nonumber \\&\quad \le C\langle t\rangle ^{-1-\delta } E_3(u(t))\le C\langle t\rangle ^{-1-\delta }A^2\varepsilon _0^2, \end{aligned}$$

(3.26)

which implies (3.21).

This completes the proof of scattering part of Theorem 1.1.

3.4 Rigidity

Assume that u is the global solution to the initial-boundary value problem (1.3), (1.8), (1.9). In the above subsection, we have shown that u will scatter, that is, there is $({\overline{u}}_0,{\overline{u}}_1)\in {\dot{H}}^1({\mathbb {R}}^{+})\times L^2({\mathbb {R}}^{+})$ such that

$$\begin{aligned} \lim _{t\rightarrow +\infty }\big (\Vert u_{\xi }-{\overline{u}}_{\xi }\Vert _{L^2_{x}({\mathbb {R}}^{+})}+\Vert u_{\eta }-{\overline{u}}_{\eta }\Vert _{L^2_{x}({\mathbb {R}}^{+})}\big )=0, \end{aligned}$$

(3.27)

where ${\overline{u}}\in C({\mathbb {R}}^{+};{\dot{H}}^1({\mathbb {R}}^{+}))\cap C^1({\mathbb {R}}^{+};L^2({\mathbb {R}}^{+}))$ is the unique global solution to the initial-boundary value problem of homogeneous linear wave equations

$$\begin{aligned} {\left\{ \begin{array}{ll} {\overline{u}}_{\xi \eta }=0, ~t>0,x>0,\\ {\overline{u}}(t,0)=0, ~~t\ge 0,\\ t=0: {\overline{u}}={\overline{u}}_0(x), {\overline{u}}_{t}={\overline{u}}_{1}(x),~ x\ge 0. \end{array}\right. } \end{aligned}$$

(3.28)

Now assume that $({\overline{u}}_0,{\overline{u}}_1)=(0,0)$, i.e., ${\overline{u}}=0$. Then by (3.27) we have

$$\begin{aligned} \lim _{t\rightarrow +\infty }E^c(u(t))=0. \end{aligned}$$

(3.29)

It follows from (3.29) that for any ${\overline{\varepsilon }}>0$, there exists $t_1=t_1({\overline{\varepsilon }})>0$, such that

$$\begin{aligned} {\mathcal {E}}^{c}(u(t_1))\le {\overline{\varepsilon }}. \end{aligned}$$

(3.30)

By Lemma 2.7 we have

$$\begin{aligned}&\sup _{0\le s\le t_1}{E}^{c}(u(s))+ {{\mathcal {E}}}^{c}(u(t_1))\nonumber \\&\quad \le C E^{c}(u(t_1))+C\int _0^{t_1}\Vert u^{T}_{\xi }F\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds+C\int _0^{t_1}\Vert u^{T}_{\eta } F\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds. \end{aligned}$$

(3.31)

By Lemma 2.3 and (3.18), we get

$$\begin{aligned} \int _0^{t_1}\Vert u^{T}_{\xi }F\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds&\le C\big \Vert |u_{\xi }|^2|u_{\eta }|\big \Vert _{L_{s,x}^1}\le C\Vert \langle \eta \rangle ^{-\frac{1+\delta }{2}} u_{\xi }\Vert _{L^2_{s,x}}^2\Vert \langle \eta \rangle ^{1+\delta }u_{\eta }\Vert _{L^{\infty }_{s,x}}\nonumber \\&\le C\sup _{0\le s\le t_1}E^{1/2}_3(u(s)){{\mathcal {E}}}^{c}(u(t_1)) \le CA\varepsilon _0{{\mathcal {E}}}^{c}(u(t_1)). \end{aligned}$$

(3.32)

Similarly,

$$\begin{aligned} \int _0^{t_1}\Vert u^{T}_{\eta }F\Vert _{L_{x}^1({\mathbb {R}}^{+})}ds&\le C\big \Vert |u_{\eta }|^2|u_{\xi }|\big \Vert _{L_{s,x}^1}\le C\Vert \langle \xi \rangle ^{-\frac{1+\delta }{2}} u_{\eta }\Vert _{L^2_{s,x}}^2\Vert \langle \xi \rangle ^{1+\delta }u_{\xi }\Vert _{L^{\infty }_{s,x}}\nonumber \\&\le C\sup _{0\le s\le t_1}E^{1/2}_3(u(s)){{\mathcal {E}}}^{c}(u(t_1)) \le CA\varepsilon _0{{\mathcal {E}}}^{c}(u(t_1)). \end{aligned}$$

(3.33)

Then it follows from (3.30), (3.31), (3.32) and (3.33) that

$$\begin{aligned}&\sup _{0\le s\le t_1}{E}^{c}(u(s))+ {{\mathcal {E}}}^{c}(u(t_1))\le C_3{\overline{\varepsilon }}+C_3A\varepsilon _0{{\mathcal {E}}}^{c}(u(t_1)). \end{aligned}$$

(3.34)

If $\varepsilon _0$ is sufficiently small such that $2C_3A\varepsilon _0\le 1$, then we get

$$\begin{aligned} \sup _{0\le s\le t_1}{E}^{c}(u(s))+ {{\mathcal {E}}}^{c}(u(t_1)) \le 2C_3{\overline{\varepsilon }}. \end{aligned}$$

(3.35)

By the arbitrariness of ${\overline{\varepsilon }}$, we have

$$\begin{aligned} \sup _{0\le s\le t_1}{E}^{c}(u(s))=0. \end{aligned}$$

(3.36)

Particularly,

$$\begin{aligned} {E}^{c}(u(0))=0, \end{aligned}$$

(3.37)

which gives

$$\begin{aligned} u(0,x)=u_0(x)=0,~~\partial _tu(0,x)=u_1(x)=0,~~x\in {\mathbb {R}}^+. \end{aligned}$$

(3.38)

By the uniqueness of global classical solution to the initial-boundary value problem (1.3), (1.8), (1.9), we get

$$\begin{aligned} u(t,x)=0,~~t\in {\mathbb {R}}^+,~x\in {\mathbb {R}}^+. \end{aligned}$$

(3.39)

Now we have completed the proof of rigidity part of Theorem 1.1.