Lower Bound of the Lifespan of the Solution to Systems of Quasi-linear Wave Equations with Multiple Propagation Speeds

TL;DR

This paper investigates the lifespan of solutions to 2D systems of quasi-linear wave equations with multiple speeds, especially when cubic nonlinearities do not satisfy the strong null-condition, using advanced energy and estimate techniques.

Contribution

It provides a lower bound on the lifespan of solutions when the strong null-condition is not met, extending understanding of solution longevity in complex wave systems.

Findings

Established a lower bound for solution lifespan

Extended analysis to cases lacking strong null-condition

Improved energy estimate methods for quasi-linear systems

Abstract

We consider the Cauchy problem of systems of quasilinear wave equations in 2-dimensional space. We assume that the propagation speeds are distinct and that the nonlinearities contain quadratic and cubic terms of the first and second order derivatives of the solution. We know that if the all quadratic and cubic terms of nonlinearities satisfy $Strong$ $Null$-$condition$, then there exists a global solution for sufficiently small initial data. In this paper, we study about the lifespan of the smooth solution, when the cubic terms in the quasi-linear nonlinearities do not satisfy the Strong null-condition. In the proof of our claim, we use the $ghost$ $weight$ energy method and the $L^{\infty}$-$L^{\infty}$ estimates of the solution, which is slightly improved.

Equations

$${\square }_i{u}^i={\partial }_0^2{u}^i-c_i^2△u^i=F^i(\partial u,{\partial }^{2}u)(x,t)\in {\mathbb{R}}^2\times (0,\infty ),$$

$$u^i(x,0)=\epsilon f^i(x),{\partial }_t{u}^i(x,0)=\epsilon g^i(x)x\in {\mathbb{R}}^2,$$

$$0<c_1<c_2<\cdots <c_m.$$

$$F^i(\partial u,{\partial }^{2}u)=\sum _{\ell =1}^m\sum _{\alpha ,\beta =0}^2{A}_{\ell }^{i,\alpha \beta }(\partial u){\partial }_{\alpha }{\partial }_{\beta }{u}^{\ell }+B^i(\partial u),$$

$$A_{\ell }^{i,\alpha \beta }(\partial u)=\sum _{j=1}^m\sum _{\gamma =0}^2{a}_{\ell j}^{i,\alpha \beta \gamma }{\partial }_{\gamma }{u}^j+\sum _{j,k=1}^m\sum _{\gamma ,\delta =0}^2{c}_{\ell jk}^{i,\alpha \beta \gamma \delta }{\partial }_{\gamma }{u}^j{\partial }_{\delta }{u}^k+O(|\partial u{|}^3)$$

$$B^i(\partial u)=\sum _{j,k=1}^m\sum _{\alpha ,\beta =0}^2{b}_{jk}^{i,\alpha \beta }{\partial }_{\alpha }{u}^j{\partial }_{\beta }{u}^k+\sum _{j,k,\ell =1}^m\sum _{\alpha ,\beta ,\gamma =0}^2{d}_{jk\ell }^{i,\alpha \beta \gamma }{\partial }_{\alpha }{u}^j{\partial }_{\beta }{u}^k{\partial }_{\gamma }{u}^{\ell }+O(|\partial u{|}^4).$$

$$A_{\ell }^{i,\alpha \beta }(v)=A_i^{\ell ,\alpha \beta }(v)=A_i^{\ell ,\beta \alpha }(v)$$

$$|A_{\ell }^{i,\alpha \beta }(v)|<\frac{(\min \{1,c_1\}{)}^2}{2m}.$$

$$u^i(x,t)=0\text{mbox}for|x|\ge c_{m}t+M.$$

$$\begin{array}{rl}{a}_{\ell j}^{i,\alpha \beta \gamma }=0& \text{mbox}when(j,\ell )\ne (i,i)\\ b_{jk}^{i,\alpha \beta }=0& \text{mbox}whenj\ne k\end{array}$$

$${\Phi }_{\ell }^i(X)=\sum _{\alpha ,\beta ,\gamma =0}^2{a}_{\ell \ell }^{i,\alpha \beta \gamma }{X}_{\alpha }{X}_{\beta }{X}_{\gamma },$$

$${\Psi }_{\ell }^i(X)=\sum _{\alpha ,\beta =0}^2{b}_{\ell \ell }^{i,\alpha \beta }{X}_{\alpha }{X}_{\beta },$$

$${\Theta }_{\ell }^i(X)=\sum _{\alpha ,\beta ,\gamma ,\delta =0}^2{c}_{\ell \ell \ell }^{i,\alpha \beta \gamma \delta }{X}_{\alpha }{X}_{\beta }{X}_{\gamma }{X}_{\delta },$$

$${\Xi }_{\ell }^i(X)=\sum _{\alpha ,\beta ,\gamma =0}^2{d}_{\ell \ell \ell }^{i,\alpha \beta \gamma }{X}_{\alpha }{X}_{\beta }{X}_{\gamma }.$$

$${\varphi }_{\ell }^i(X)\equiv 0\text{mbox}for{X}_0^2=c_{\ell }^2(X_1^2+X_2^2)$$

$${\square }_i{u}_0^i=0(x,t)\in {\mathbb{R}}^2\times (0,\infty ),$$

$$u_0^i(x,0)=f^i(x),{\partial }_0{u}_0^i(x,0)=g^i(x)x\in {\mathbb{R}}^2.$$

$${\mathcal{F}}^i(\rho ,\omega )=\underset{r\to \infty }{\lim }{r}^{\frac{1}{2}}{u}_0^i(x,t)$$

$${\mathcal{F}}^i(\rho ,\omega )=\frac{1}{2\sqrt{2}}{\int }_{\rho }^{\infty }\frac{1}{\sqrt{s-\rho }}(R_{g^i}(s,\omega )-{\partial }_s{R}_{f^i}(s,\omega ))ds,$$

$$R_h(s,\omega )={\int }_{\omega \cdot y=s}h(y)d{S}_y$$

$${\mathcal{F}}^i(\rho ,\omega )=0\text{mbox}for\rho \ge M,$$

$$\displaystyle\big{|}\partial_{\rho}^{\ell}\mathcal{F}^{i}(\rho,\omega)\big{|}\leq\frac{C}{(1+|\rho|)^{\frac{1}{2}+\ell}}$$

$$\displaystyle\big{|}r^{\frac{1}{2}}\partial_{0}^{\ell}u_{0}^{i}(r\omega,t)-(-c_{i})^{\ell}\partial_{\rho}^{\ell}\mathcal{F}^{i}(r-c_{i}t,\omega)\big{|}\leqq\frac{C(1+|r-c_{i}t|)^{\frac{1}{2}}}{t}\qquad$$

$$H_i=\underset{\begin{array}{c}\rho \in \mathbb{R}\\ \omega \in S^1\end{array}}{\max }\left\{-\frac{1}{c_i^2}{\Theta }_i^i(-c_i,\omega ){\partial }_{\rho }{\mathcal{F}}^i(\rho ,\omega ){\partial }_{\rho }^2{\mathcal{F}}^i(\rho ,\omega )\right\}$$

$$H=\max \{H_1,H_2,\cdots ,H_m\}.$$

$$\underset{\epsilon \to 0}{\mathrm{liminf}}{\epsilon }^2\log (1+T_{\epsilon })\geqq \frac{1}{H}.$$

$${\epsilon }^2\log (1+T_{\epsilon })\ge B$$

$$\Omega =x_1{\partial }_2-x_2{\partial }_1,S=t{\partial }_0+x_1{\partial }_1+x_2{\partial }_2$$

$$\Gamma =({\Gamma }_0,{\Gamma }_1,{\Gamma }_2,{\Gamma }_3,{\Gamma }_4)=({\partial }_0,{\partial }_1,{\partial }_2,\Omega ,S)$$

$${\Gamma }^a={\Gamma }_0^{a_0}{\Gamma }_1^{a_1}{\Gamma }_2^{a_2}{\Gamma }_3^{a_3}{\Gamma }_4^{a_4}$$

$$[{\Gamma }_{\alpha },{\square }_i]=-2{\delta }_{\alpha 4}{\square }_i(\alpha =0,1,2,3,4,i=1,2,\cdots ,m)$$

$$[{\partial }_{\alpha },{\partial }_{\beta }]=0(\alpha ,\beta =0,1,2),[S,{\partial }_{\alpha }]=-{\partial }_{\alpha }(\alpha =0,1,2)$$

$$[\Omega ,{\partial }_1]=-{\partial }_2,[\Omega ,{\partial }_2]={\partial }_1,[\Omega ,{\partial }_0]=0,[S,\Omega ]=0.$$

Full text

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2010 ASC: 35L05, 35L15, 35L51, 35L70.

Keywords: Lifespan, Wave Eqation, Null-condition.

1 Intrduction

In this paper, we study the Cauchy problem;

[TABLE]

where $i=1,2,\cdots,m$ and $u(x,t)=(u^{1}(x,t),u^{2}(x,t),\cdots,u^{m}(x,t))$. We denote $\partial u=(\partial_{\alpha}u)_{\alpha=0,1,2}$ with $\partial_{0}=\partial_{t}=\partial/\partial t$, $\partial_{j}=\partial/\partial x_{j}\ (j=1,2)$ and $\partial^{2}u=(\partial_{\alpha}\partial_{\beta}u)_{\alpha,\beta=0,1,2}$. Let $\varepsilon>0$ is a small parameter and assume $f^{i}(x),g^{i}(x)\in C^{\infty}_{0}(\mathbb{R}^{2})$ and $\mbox{supp}\{f^{i}\},\mbox{supp}\{g^{i}\}\subset\{x\in\mathbb{R}^{2}\ :\ |x|\leq M\}$ for some positive constant $M$. We also assume that the propagation speeds of (1.1) are distinct constants, namely we assume

[TABLE]

Each nonlinearity $F^{i}(v,w)$ is smooth near the origin and is expressed as

[TABLE]

where

[TABLE]

and

[TABLE]

Here $a_{\ell j}^{i,\alpha\beta\gamma},b_{jk}^{i,\alpha\beta},c_{\ell jk}^{i,\alpha\beta\gamma\delta},d_{jk\ell}^{i,\alpha\beta\gamma}$ are constants.

In order to derive energy estimate, we need to assume that for each $i,\ell=1,2,\cdots,m$ and $\alpha,\beta=0,1,2$,

[TABLE]

and

[TABLE]

The assumption (1.8) constitutes no additional restriction, since we will only deal with small solutions. Note that by (1.3) and (1.4), we have for any $i=1,2,\cdots,m$,

[TABLE]

For the proof of (1.9), see Theorem 4a in F. John [9]. Furthermore, in order to derive the $ghost$ $weight$ energy method, we need to assume that

[TABLE]

for each $i=1,2,\cdots,m$ and $\alpha,\beta,\gamma=0,1,2$. This assumption means that only the terms $\partial u^{i}\partial^{2}u^{i}$ and $\partial u^{j}\partial u^{j}$ $(j=1,\cdots,m)$ appear in the quadratic terms of $F^{i}$.

Our purpose of this paper is to show a precise estimate for the $lifespan$ $T_{\varepsilon}$. Here, we define $T_{\varepsilon}$ by the supremum of $T>0$ for which there exists a solution $u$ to the Cauchy problem (1.1) and (1.2) in \big{(}C^{\infty}(\mathbb{R}^{2}\times[0,T))\big{)}^{m}. To state the known results and our our result, we introduce some notations. Firstly, for $X=(X_{0},X_{1},X_{2})\in\mathbb{R}^{3}$, we define $\Phi(X)=(\Phi_{\ell}^{i}(X))_{i,\ell=1,2,\cdots,m}$, $\Psi(X)=(\Psi_{\ell}^{i}(X))_{i,\ell=1,2,\cdots,m}$, $\Theta(X)=(\Theta_{\ell}^{i}(X))_{i,\ell=1,2,\cdots,m}$ and $\Xi(X)=(\Xi_{\ell}^{i}(X))_{i,\ell=1,2,\cdots,m}$ by

[TABLE]

Moreover, let $\phi(X)=(\phi_{\ell}^{i}(X))_{i,\ell=1,2,\cdots,m}$ be a function of $X=(X_{0},X_{1},X_{2})$. If

[TABLE]

holds for each $i,\ell=1,2,\cdots,m$, then we denote $\phi\approx 0$ and we say that $\phi$ satisfies $Strong$ $Null$-$condition$. On the other hand, if (1.15) holds when $\ell=i$ $(i=1,2,\cdots,m)$, then we denote $\phi\sim 0$ and we say that $\phi$ satisfies $Standard$ $Null$-$condition$. In [5], the author showed that $\displaystyle{\liminf_{\varepsilon\to 0}\varepsilon\sqrt{T_{\varepsilon}}\geq C}$ holds for a certain positive constant $C$, provided $B^{i}(\partial u)\equiv 0$. On the other hand, the author also showed in [6] that $T_{\varepsilon}=\infty$ for sufficiently small $\varepsilon>0$, provided $\Phi$, $\Psi$, $\Theta$ and $\Xi$ satisfy Strong Null-Condition and $A_{\ell}^{i,\alpha\beta}(\partial u)\equiv 0$ holds for $\ell\neq i$. In this paper, we consider the case that $\Phi$ and $\Psi$ satisfy Strong Null-condition and $\Xi$ satisfies Standard Null-condition. Namely, we assume $\Phi\approx 0$, $\Psi\approx 0$ and $\Xi\sim 0$.

Secondly, we introduce the Friedlander radiation field $\mathcal{F}^{i}(\rho,\omega)$. Let $u^{i}_{0}(x,t)$ be the solution to the Cauchy problem of the homogeneous linear wave equation;

[TABLE]

Then we define $\mathcal{F}^{i}$ by

[TABLE]

with $x=r\omega\ (\omega\in S^{1})$ and $\rho=r-c_{i}t$. We know that $\mathcal{F}^{i}(\rho,\omega)$ is expressed by

[TABLE]

where $R_{h}(s,\omega)$ is the Radon transform of $h\in C_{0}^{\infty}(\mathbb{R}^{2})$, $i.e.$,

[TABLE]

for $s\in\mathbb{R},\ \omega\in S^{1}$. Note that $\mathcal{F}^{i}(\rho,\omega)$ satisfies

[TABLE]

[TABLE]

and

[TABLE]

for $t\geq r/(2c_{i})\geq 1$ and $\ell=1,2$. For the details about (1.19), (1.20) and (1.21), see L. Hörmander [2].

Then we define a constant

[TABLE]

and set

[TABLE]

Note that by (1.19) and (1.20), each $H_{i}$ is well-defined and nonnegative.

Now, we state our main result.

Theorem 1.1

Assume that (1.7), (1.8) and (1.10) hold for the Cauchy problem (1.1) and (1.2). Also assume $\Phi\approx 0$, $\Psi\approx 0$ and $\Xi\sim 0$. Then, if $H>0$, we have

[TABLE]

Note that the author showed the same estimate when $a_{\ell j}^{i,\alpha\beta,\gamma}=b_{jk}^{i,\alpha\beta}=0$ for all $\alpha,\beta,\gamma=0,1,2$ and $i,j,k=1,2,\cdots,m$ in [4]. Hence, our result (1.24) is a generalization of the result in [4]. Also note that we can not improve the estimate (1.24), in general, since the counter result has been shown when $m=1$ and $B^{1}(\partial u)\equiv 0$ in [3].

In the following sections, we aim at showing (1.24). In section 2, we prepare some notations and state a lemma which implies (1.24). We also discus about the estimates of the null-form. In section 3, we will show the $L^{\infty}$-$L^{\infty}$ estimates of solutions to the wave equation. It is an improvement of the one showed in [8]. In section 4, we concentrate to show $a$ $priori$ estimates of the solution, by using the ghost energy inequality and the method of ordinary differential equation along the characteristic curves.

2 Preliminary for the proof of Theorem 1.1

Our main theorem is immediately derived from the following lemma.

Lemma 2.1

Under the same situation as Theorem 1.1, choose a positive constant $B$ to be $B<1/H$. Then there exists a constnat $\varepsilon_{0}(B)>0$ such that

[TABLE]

holds for $0<\varepsilon<\varepsilon_{0}(B)$.

In order to state another lemma which causes Lemma 2.1, we introduce some notations. At first, we introduce the following differential operators,

[TABLE]

and denote

[TABLE]

and

[TABLE]

for a multi-index $a=(a_{0},a_{1},a_{2},a_{3},a_{4})$. We can verify the following commutator relations;

[TABLE]

and

[TABLE]

Here, $[A,B]=AB-BA$ and $\delta_{\alpha\beta}$ is the Kronecker delta.

Secondly, we define norms. Let $v(x,t)=(v^{1}(x,t),v^{2}(x,t),\cdots,v^{m}(x,t))$ be a vector valued function defined on $\mathbb{R}^{2}\times[0,T)$, then we set

[TABLE]

where $k$ is a nonnegative integer and $|a|=a_{0}+a_{1}+\cdots+a_{4}$ for a multi-index $a=(a_{0},a_{1},a_{2},a_{3},a_{4})$.

Then, we find that the following lemma implies Lemma 2.1.

Lemma 2.2

Let $u(x,t)=(u^{1}(x,t),u^{2}(x,t),\cdots,u^{m}(x,t))\in(C^{\infty}(\mathbb{R}^{2}\times[0,T)))^{m}$ be a solution to (1.1) and (1.2). Choose an integer $k$ so that $k\geq 21$. Let $B>0$ be a constant so that $B<1/H$ and also let $J>0$ be a constant. Then, there exist constants $K=K(B)>0$ and $\varepsilon_{0}=\varepsilon_{0}(J,B)>0$ such that, if

[TABLE]

holds for $0<\varepsilon<\varepsilon_{0}$, then

[TABLE]

holds for the same $\varepsilon$. Here, we have set $T_{B}=\min\{T,t_{B}\}$ and $t_{B}=\exp(B/\varepsilon^{2})-1$.

Proof of Lemma 2.1 providing Lemma 2.2: We show that Lemma 2.2 implies Lemma 2.1 by contradiction. If the statement of Lemma 2.1 is incorrect, there exists a positive constant $B_{0}(<1/H)$ such that for any $\varepsilon>0$, there exists $\delta=\delta(\varepsilon)>0$ such that

[TABLE]

On the other hand, by the local existence theorem which was shown in A. Majda [12], we find that there are positive constants $\varepsilon_{1}$ and $t_{\varepsilon}$ such that there exists a smooth solution $u(x,t)\in C^{\infty}(\mathbb{R}^{2}\times[0,t_{\varepsilon}))$ to (1.1) and (1.2) for $0<\varepsilon<\varepsilon_{1}$. Let $L>0$ be a constant satisfying $[\partial u(0)]_{k}+\langle u(0)\rangle_{k+1}\leq L\varepsilon$ and set $J_{0}=2\max\{K(B_{0}),L\}$, where $K(B_{0})$ is the constant determined in Lemma 2.2 with $B=B_{0}$. Then we can define a positive constant $\tau_{\varepsilon}$ by

[TABLE]

for each $\varepsilon\in(0,\varepsilon_{*})$. Here we have set $\varepsilon_{*}=\min\{\varepsilon_{0}(J_{0},B_{0}),\varepsilon_{1}\}$. Note that (2.2) holds for $\varepsilon\in(0,\varepsilon_{*})$ with $J=J_{0}$ and $T=\tau_{\varepsilon}$. Moreover, by using (1.7) and (1.8), we can show $\tau_{{\varepsilon}}<T_{{\varepsilon}}$ for each $\varepsilon\in(0,\varepsilon_{*})$. (For the detail, see the proof of Lemma 2.1 in [5].) This means that

[TABLE]

holds for $\varepsilon\in(0,\varepsilon_{*})$. However, as mentioned above, there exists a constant $\delta=\delta(\varepsilon_{*})$ such that (2.4) holds. In that case, we find that $T_{B_{0}}=\min\{\tau_{\delta},t_{B_{0}}\}=\tau_{\delta}$ and hence that Lemma 2.2 implies

[TABLE]

This contradicts to (2.5) and therefore we find that the claim of Lemma 2.1 is correct.

In the rest of this paper, we aim at showing Lemma 2.2. For this purpose, we prepare a proposition with respect to the $null$-$form$. Set $\displaystyle{c_{*}=\min_{1\leq i\leq m}\{c_{i}-c_{i-1}\}/3}$ with $c_{0}=0$. We see $c_{*}>0$ from (1.3). Also we set

[TABLE]

and

[TABLE]

We find that $\Lambda_{i}(T)\cap\Lambda_{j}(T)=\emptyset$ holds for any $T>0$, if $i\neq j$ and that there exists a constant $C_{1}>0$ such that

[TABLE]

holds for any $T>0$, if $i\neq j$.

In order to derive a good decay property from the null-form in $\Lambda_{i}(T)$, we introduce the following operators;

[TABLE]

Then we find that

[TABLE]

and hence that

[TABLE]

Now we have the following.

Proposition 2.1

Let $T>1$ be a constant and let $k$ be a positive integer. Let $v(x,t)=(v^{1}(x,t),v^{2}(x,t),\cdots,v^{m}(x,t))$ and $w(x,t)=(w^{1}(x,t),w^{2}(x,t),\cdots,w^{m}(x,t))$ be functions belonging to $(C^{\infty}(\mathbb{R}^{2}\times[0,T)))^{m}$. Assume that $\Phi\approx 0$, $\Psi\approx 0$, $\Xi\sim 0$ and (1.10) hold. Then, there exists a positive constant $C_{2}$ independent of $T$ such that

[TABLE]

[TABLE]

[TABLE]

and especially

[TABLE]

hold for $i,j=1,2,\cdots,m$. Moreover, we find that

[TABLE]

holds for $(x,t)\in\Lambda_{j}(T)\cap\{(y,s):s\geq 1\}$, $i,j=1,2,\cdots,m$, and that

[TABLE]

[TABLE]

and

[TABLE]

hold for $(x,t)\in\Lambda_{i}(T)\cap\{(y,s):s\geq 1\}$, $i=1,2,\cdots,m$. Here we have set

[TABLE]

For the proof of Proposition 2.1, see Proposition 2.1 in [6].

3 $L^{\infty}$-$L^{\infty}$ estimate

In this section, we will show a weighted $L^{\infty}$-$L^{\infty}$ estimate of solutions to inhomogeneous wave equations. It is an improvement of the estimate in Proposition 4.2 in [8]. Let $c$ and $T$ be positive constants and $F$ be a function in $C^{\infty}(\mathbb{R}^{2}\times[0,T))$. Then we introduce an operator $L_{c}(F)$;

[TABLE]

for $(x,t)\in\mathbb{R}^{2}\times[0,T)$. We know that $L_{c}(F)$ is the solution to the Cauchy problem;

[TABLE]

Then we have the following.

Proposition 3.1

Let $c_{i}$ $(i=1,2,\cdots,m)$ be the propagation speeds defined in (1.3). Let $T>0$ and $F,G,H\in C^{\infty}(\mathbb{R}^{2}\times[0,T))$. Choose $\mu>0$, $\nu>0$ and $\rho>0$ arbitrarily. Then, there exist positive constants $\tilde{C}_{\mu}$, $\hat{C}_{\nu}$ and $\dot{C}_{\rho}$ independent of $T$ such that

[TABLE]

and

[TABLE]

hold for $(x,t)\in{\bf R}^{2}\times[0,T)$. Here $\nabla=(\partial_{1},\partial_{2})$ and we have set

[TABLE]

[TABLE]

[TABLE]

Proof of Proposition 3.1:    By the same argument with the proof of Propositions 4.1 and 4.2 in [8], we obtain (3.2) and (3.3) when $H(x,t)\equiv 0$. Therefore, we have only to show (3.3) when $G(x,t)\equiv 0$. Without loss of generality, we may assume $c_{i}=1$ and for the sake of simplicity, we denote the constant depending on $\rho$ by $C$ which may change line by line, during this section. Set

[TABLE]

with $r=|x|$ and define

[TABLE]

then we have

[TABLE]

Firstly, we deal with $P_{1}(\partial_{\ell}H)(x,t)$. Following the computation made in Section 4 of [8], we find

[TABLE]

where we have set

[TABLE]

Here we have used the following notation:

[TABLE]

with $\lambda_{-}=|t-s-r|$, $\lambda_{+}=t-s+r$ and $\delta=\min\{r,1/2\}$. Thus we aim to show

[TABLE]

In oder to show (3.4), we use the following estimates which are proved in Lemma 4.1 in [7].

Lemma 3.1

Let $(\lambda,s)\in D_{1}\cup D_{2}$. then we have

[TABLE]

where, $h(p)=1$ for $p>0$ and $h(p)=0$ for $p\leq 0$.

First we evaluate $I_{1}$. When $t-r-s>0$ and $\lambda>\lambda_{+}-\delta$, we have

[TABLE]

since $\lambda-\lambda_{-}>r$. Moreover, we find that

[TABLE]

Hence, by (3.5), we get

[TABLE]

where we have set

[TABLE]

It follows that

[TABLE]

When we deal with $B_{1,j}$, we may assume $t>r$, since $B_{1,j}=0$ if $t\leq r$. Integrating by parts, we find

[TABLE]

where we have used $0<\delta<1/2$ and the facts

[TABLE]

Hence we have

[TABLE]

When $s>(t-r)_{+}$, we have

[TABLE]

which imply

[TABLE]

Therefore, we get

[TABLE]

[TABLE]

Summing up (3.8), (3), (3.10), (3.11) and (3.12), we obtain (3.4) for $k=1$, since $\delta/r\leq 2/(1+r)$.

In the remainder of the proof of (3), we assume $r\geq 1/2$ so that $\delta=1/2$, because $D_{2}$ is the empty set, if $0<r<1/2$.

Since $\lambda=\lambda_{-}+1/2$ or $\lambda=\lambda_{+}-1/2$ for $(\lambda,s)\in D_{2}^{\prime}$, we obtain (3.4) for $k=2$ analogously to the previous argument.

Next we evaluate $I_{3}$. Note that $\lambda>\lambda_{-}+1/2$ for $(\lambda,s)\in D_{2}$ and that

[TABLE]

for $\lambda>\lambda_{-}+1/2$. Therefore we get from (3.5)

[TABLE]

where we have set

[TABLE]

When $1\leq j\leq m$, changing variables by

[TABLE]

we have

[TABLE]

where

[TABLE]

On the other hand, when $j=0$, we have

[TABLE]

Therefore we have (3.4) for $k=3$.

Next we evaluate $I_{4}$. Since $\lambda+s+r-t\geq 1/2$ for $\lambda\geq\lambda_{-}+1/2$, we get from (3.6)

[TABLE]

which yields (3.4) for $k=4$.

Next we evaluate $I_{5}$. It follows from $\lambda_{-}+1/2\leq\lambda\leq\lambda_{+}-1/2$ that

[TABLE]

Hence we get from (3.7)

[TABLE]

where we have set

[TABLE]

Changing variables by (3.13), we have

[TABLE]

Changing variables by (3.13) and then by $\sigma=\psi_{j}(\alpha,\beta)$, we get

[TABLE]

where

[TABLE]

It has been shown in Lemma 3.13 in [11] that

[TABLE]

Therefore, if $j\neq i$, we have

[TABLE]

On the other hand, if $j=i$, since $\gamma_{i}=r-t$, we have

[TABLE]

Since we can deal with $C_{5,j}$ similarly to $B_{5,j}$, we obtain (3.4) for $k=5$.

Secondly, we deal with $P_{2}(\partial_{\ell}H)$ We can assume $t>r$, since otherwise $E_{2}$ is empty. Switching to polar coordinates,

[TABLE]

we get

[TABLE]

By Proposition 5.2 in [1], we have

[TABLE]

for $0<s<t-r$ and $0<\lambda<\lambda_{-}$. It follows from the fact

[TABLE]

that

[TABLE]

where we have set

[TABLE]

Changing variables by (3.13), we get

[TABLE]

Here

[TABLE]

implies

[TABLE]

Thus, combining (3.17) and (3.18), we obtain

[TABLE]

Thirdly, we deal with $P_{3}(\partial_{\ell}H)$. We can assume $t>r+1/2$, since otherwise $E_{3}$ is empty. Integrating by parts in $y$ and switching to polar coordinates as in (3.15), we get

[TABLE]

where we have set

[TABLE]

We see from (3.15) and (3.16) that

[TABLE]

As shown in the proof of Proposition 4.2 in [7], we have

[TABLE]

Therefore, since $r+1\leq 2(\lambda_{+}-\lambda)$ for $\lambda<t-r-s-1/2$, we get from (3.13)

[TABLE]

This completes the proof of (3.3).

4 $a$ $priori$ estimates

In this section, we derive the $a$ $priori$ estimate (2.3) assuming (2.2). For this purpose, we introduce a notation. Let the assumptions of Lemma 2.1 be fulfilled and let $p(x,t)$ and $q(x,t)$ be functions defined on a set $D\subset\mathbb{R}^{2}\times[0,T)$. Then we denote

[TABLE]

when there exist constants $K=K(B)>0$ and $\varepsilon_{0}=\varepsilon_{0}(J,B)>0$ such that, if (2.2) holds for $0<\varepsilon<\varepsilon_{0}$, then

[TABLE]

for the same $\varepsilon$. We can easily show that if $p_{1}(x,t)=O^{*}(q(x,t))$ and $p_{2}(x,t)=O^{*}(q(x,t))$, then $p_{1}(x,t)+p_{2}(x,t)=O^{*}(q(x,t))$. Then our task to prove Lemma 2.1 is showing

[TABLE]

Also we will express constants determined independently of $J$ and $T$ by $K_{n}\ (n\in\mathbb{N})$ in the following argument.

Now we aim to show (4.1). By (3.1), we can write

[TABLE]

where $u^{i}_{0}(x,t)$ is the solution to (1.15), (1.16) and satisfies for any nonnegative integer $p$,

[TABLE]

with some constant $C_{0}=C_{0}(f^{i},g^{i},p)>0$. Then, we have for a multi-index $a=(a_{0},a_{1},\cdots,a_{4})$,

[TABLE]

with some constants $C_{a,b}$. Here, $v^{i}_{a}=v_{a}^{i}(x,t)$ is the solution to the Cauchy problem;

[TABLE]

with functions $\phi^{i}_{a},\ \psi_{a}^{i}\in C_{0}^{\infty}(\mathbb{R}^{2})$ determined by $(f^{j},g^{j})_{j=1,2,\cdots,m}$ suitably. Indeed, by the commutation relations of $\Gamma_{\alpha}$ and $\Box_{i}$ and by the definition of $L_{c_{i}}(F^{i})$, we have

[TABLE]

and

[TABLE]

Since $F^{i}$ is quadratic, we can denote $F^{i}(x,0)=\varepsilon^{2}\psi^{i}(x)\in C^{\infty}_{0}(\mathbb{R}^{2})$. Hence we have

[TABLE]

where $v^{i}=v^{i}(x,t)$ is the solution to the Cauchy problem;

[TABLE]

This implies (4.4) when $|a|=1$. Repeating the above argument, we can obtain (4.4) for any $a$.

Note that, as with (4.3), we have for any nonnegative integer $p$,

[TABLE]

with some constant $C^{\prime}_{0}=C^{\prime}_{0}(f^{1},\cdots,f^{m},g^{1},\cdots,g^{m},p)>0$. It follows from (4.2) and (4.4) that

[TABLE]

Therefore, our task for the proof of (4.1) is to show

[TABLE]

We will show (4.9) by dividing the area into some parts.

Firstly, we assume $\displaystyle{0\leq t\leq 1/{\varepsilon}}$. In this region, we can show the sharper estimates;

[TABLE]

For this purpose, we prepare two propositions with respect to the energy.

Proposition 4.1

Let $u(x,t)\in C^{2}(\mathbb{R}^{2}\times[0,T))$ be a function satisfying $||u||_{2,T}<\infty$. Then, there exists a constant $C_{3}>0$ such that

[TABLE]

holds for $(x,t)\in\mathbb{R}^{2}\times[0,T)$.

Proposition 4.2

Let $\displaystyle{u(x,t)=(u^{1}(x,t),u^{2}(x,t),\cdots,u^{m}(x,t))}$ $\displaystyle{\in(C^{\infty}(\mathbb{R}^{2}\times[0,T)))^{m}}$ be the solution to (1.1) and (1.2) and also let $\ell$ be a positive integer. Assume (1.7). Then there exist constants $\delta>0$ and $C_{4}=C_{4}(\ell)>0$ such that if $\displaystyle{|\partial u|_{[\frac{\ell+1}{2}],T}<\delta}$ holds, then

[TABLE]

holds for $0\leq t<T$.

We omit the proof of the propositions. For the details of Proposition 4.1, see [10]. On the other hand, we get Proposition 4.2 by the usual energy argument for the quasilinear wave equations.

By (2.2) and $k\geq 21$, we have $|\partial u|_{[\frac{k+5}{2}],\frac{1}{\varepsilon}}\leq|\partial u|_{k,\frac{1}{\varepsilon}}\leq J\varepsilon<\delta$ for $0<\varepsilon<\varepsilon_{0}$, if we take $\varepsilon_{0}$ to be $J\varepsilon_{0}<\delta$. Hence, by (2.2) and (4.12), we have

[TABLE]

if we take $\varepsilon_{0}$ to be $\varepsilon_{0}\leq 1$ and $J^{2}\varepsilon_{0}\leq 1$. Therefore, by (1.9), (2.2), (3.2), (4.11) and (4), we have for $|c|\leq k+1$

[TABLE]

if we take $\mu$ and $\varepsilon_{0}$ to be $\mu<1/{16}$ and $J^{8}\varepsilon_{0}\leq 1$. As for $\partial L_{c_{i}}(\Gamma^{b}F^{i})$ with $|b|\leq k$, when $0\leq t\leq 1$, it follows from (1.9) that $1+|x|+t$ and $1+||x|-c_{i}t|$ are bounded in the support of the solution $u^{i}(x,t)$. Hence, by (4.4), (4.7) and (4), we find

[TABLE]

On the other hand, by (1.9), (2.2), (3.3) with $G=0$, (4.11) and (4), we have

[TABLE]

if we take $\nu$ and $\varepsilon_{0}$ to be $\nu<1/{16}$ and $J^{8}\varepsilon_{0}<1$. Therefore, when $1\leq t$, by (1.9), (4.4), (4.7) , (4), (4) and the identity

[TABLE]

we have

[TABLE]

Therefore we obtain (4.10).

Secondly, we assume $\displaystyle{1/{\varepsilon}\leq t\leq T_{B}}$. In this region, we need more precise energy estimate:

Proposition 4.3

Let $u(x,t)=(u^{1}(x,t),u^{2}(x,t),\cdots,u^{m}(x,t))\in(C^{\infty}(\mathbb{R}^{2}\times[0,T))^{m}$ be the solution to (1.1) and (1.2) under the same assumption in Theorem 1.1. Also let $\ell$ be a positive integer. Then there exist positive constants $C_{5}$ and $\delta$ such that if

[TABLE]

holds, then

[TABLE]

holds for $0\leq t_{0}\leq t<T$. Here, we have set

[TABLE]

For the proof of (4.19), see Proposition 4.1 in [6], in there we used the $ghost$ $weight$ energy method.

Remark: The difference of situations between Proposition 4.1 in [6] and Proposition 4.3 in this paper is the power of $1+||x|-c_{i}t|$ in $[\partial u^{i}]_{k}$ which is the norm we are going to estimate. To derive the $ghost$ $weight$ energy method, we need to suppose that $|\partial u^{i}(x,t)|_{k}$ is equivalent to $1+||x|-c_{i}t|$ in the area where $|x|$ is small. Thus we need to define another norm $[[[\partial u]]]_{k}$ and assume (4.18) in Proposition 4.3.

In order to use (4.19) with $\ell=k+9$ and $T=T_{B}$, we also show that

[TABLE]

holds. By (2.2) and $k\geq 21$, we find that

[TABLE]

if we take $\varepsilon_{0}$ to be $J^{2}\varepsilon_{0}\leq 1$. Furthermore, if we obtain

[TABLE]

then by (1.9), (2.2), (4.4), (4.16) and (4.25), we find that

[TABLE]

if we take $\varepsilon_{0}$ to be $J^{2}\varepsilon_{0}\leq 1$. Hence, by (4.3), (4.7), (4.10), (4.25) and (4.26), we have (4.23).

In order to prove (4.9) and (4.25), we show that for any positive integer $\ell\leq k+1$ and for any positive constant $\eta$

[TABLE]

and

[TABLE]

hold. We will show (4.27) and (4.28) step by step.

At first, by (1.5), (1.6), (1.9), (2.17), (2.18), (3.2), (3.3), (4.11) and $\varepsilon^{2}\log(1+t)\leq B$, we have for any $\mu_{1}>0$ and $\rho_{1}>0$,

[TABLE]

and

[TABLE]

[TABLE]

where we have set

[TABLE]

Next, we estimate $(1+s+|y|)^{-\frac{7}{16}+\mu_{1}}([[\partial u(y,s)]]_{\ell+2}+\langle\langle u(y,s)\rangle\rangle_{\ell+3})$ for $(y,s)\in\mathbb{R}^{2}\times[0,t]$. By the same manner as (4.29), for any $\mu_{2}>0$, we obtain by (1.5), (1.6), (1.9), (2.17), (2.18), (3.2), (3.3), (4.3), (4.7), (4.8) and (4.17)

[TABLE]

[TABLE]

Moreover, by the same manner as (4), for any $\mu_{3}>0$ we obtain (1.5), (1.6), (1.9), (3.2), (3.3), (4.3), (4.7), (4.8) and (4.17)

[TABLE]

if we take $\mu_{1},\mu_{2},\mu_{3}$ to be $\mu_{1}+\mu_{2}+\mu_{3}<5/{16}$. Combining (4.29), (4), (4) and (4) and taking $\mu_{1}=\eta$, $\varepsilon_{0}\leq 1$, $J\varepsilon_{0}\leq 1$, we have (4.27) and (4.28).

Now we show (4.25). It follows from (2.2), (4.28) and $k\geq 21$ that

[TABLE]

if we take $\varepsilon_{0}$ to be $J^{6}\varepsilon_{0}\leq 1$. This implies (4.25) and therefore (4.23). Furthermore, (4.23) implies that there exists a positive constant $K_{1}$ such that

[TABLE]

holds for $0<\varepsilon<\varepsilon_{0}$. Hence, by (4.19) and (4.33), we have

[TABLE]

Therefore , by (4.11), (4.27), (4.28) and (4), we obtain

[TABLE]

and

[TABLE]

if we choose $\eta$ and $\varepsilon_{0}$ to be $0<\eta+C_{5}K_{1}\varepsilon_{0}^{\frac{1}{2}}<1/{16}$, $0<C_{5}K_{1}\varepsilon^{\frac{1}{2}}_{0}<1/256$ and $J^{256}\varepsilon_{0}\leq 1$. Hence, by (4.3), (4.7) and (4), we have

[TABLE]

Hence, by (4.37),we obtain

[TABLE]

and therefore by (4.3), (4.7), (4.8), (4.17), (4) and (4.38), we have

[TABLE]

Note that (4.38) and (4.39) are stronger than we needed with respect to the order of derivatives. We will make use of the strength of the estimates below.

On the other hand, in order to estimate $\partial u$, we introduce a subset of $\mathbb{R}^{2}\times[0,T_{B})$ by

[TABLE]

and discuss by dividing the area $\mathbb{R}^{2}\times[0,T_{B})$ into out-side and in-side of $\tilde{\Lambda}_{i}(T_{B})$. We also introduce notations;

[TABLE]

and

[TABLE]

Then we find that

[TABLE]

holds for some constant $C_{6}>0$ and that

[TABLE]

holds for sufficiently small $\varepsilon>0$. Hence, it follows from (4) and (4.40) that

[TABLE]

for $i=1,2,\cdots,m$. Hence, by (4.3), (4.7), (4.8), (4.16), (4.38) and (4), we obtain

[TABLE]

for $i=1,2,\cdots,m$. Especially, by (4.2), (4.3), (4.4), (4.7), (4.16), (4.38) and (4), we obtain

[TABLE]

and

[TABLE]

Now, the task left for us is to show

[TABLE]

for $i=1,2,\cdots,m$. We use the method of ordinary differential equation along the pseudo characteristic curves. Let $u(x,t)=(u^{1}(x,t),u^{2}(x,t),\cdots,u^{m}(x,t))$ be the solution to (1.1) and (1.2) and denote $x=r\omega,\ (r=|x|,\ \omega\in S^{1})$. Then, for fixed $\lambda\in\mathbb{R}$ and $\omega\in S^{1}$, we define the $i$-th pseudo characteristic curve in $(r,t)$-plane by the solution $r=r^{i}(t;\lambda)$ of the Cauchy problem;

[TABLE]

where $t_{0}=1/{\varepsilon}$ when $|\lambda|<\varepsilon^{-\frac{1}{4}}$, and $t_{0}=\lambda^{4}$ when $|\lambda|\geq\varepsilon^{-\frac{1}{4}}$. Namely, the initial point $(r^{i}(t_{0};\lambda)\omega,t_{0})$ is on $\partial\tilde{\Lambda}_{i}(T_{B})$ for each $\lambda\in\mathbb{R}$ and $\omega\in S^{1}$. Denote

[TABLE]

then we find that

[TABLE]

holds for each $i=1,2,\cdots,m$. For the details, see [5]. Now, we can transform the equation (1.1) into an ordinary differential equation along the pseudo characteristic curve. For a vector valued function $v=(v^{1},v^{2},\cdots,v^{m})$, set

[TABLE]

then we obtain an identity

[TABLE]

Note that the differential operator $\partial_{0}+\kappa_{i}\partial_{r}$ in the left hand side of (4) means the derivative along $r=r^{i}(t;\lambda)$ in $(r,t)$-plane. Furthermore, by (2.10), (2.11) and the definition of $\tilde{\Lambda}_{i}(T)$, we have for any $\alpha,\beta=0,1,2$ and $i=1,2,\cdots,m$

[TABLE]

in $\tilde{\Lambda}_{i}(T_{B})$. By (2.2), (2.18) and (4.49), we have

[TABLE]

if we take $\varepsilon_{0}$ to be $J\varepsilon_{0}\leq 1$. Therefore, it follows from (4), (4.49) and (4) that

[TABLE]

Now we show (4.46) by induction with respect to $k$. Choose a point $(x,t)\in\tilde{\Lambda}_{i}(T_{B})$, then there exist $\lambda\in\mathbb{R}$ and $\omega\in S^{1}$ such that $x=r\omega$ and $(r\omega,t)\in\mathcal{J}^{i}(\lambda;\omega)$. At first, we show

[TABLE]

for $i=1,2,\cdots,m$. Setting $v=u$ in (4.51), we have by (1.1), (2.2), (2.17) and (2.19),

[TABLE]

if we take $\varepsilon_{0}$ to be $J\varepsilon_{0}\leq 1$. Integrating (4.52) from $t_{0}$ to $t$, we have

[TABLE]

which implies

[TABLE]

if we take $\varepsilon_{0}$ to be $J^{4}\varepsilon_{0}\leq 1$. Moreover, integrating (4.47) and using (4.55), we have

[TABLE]

Hence, by (4), (4.54) and (4), we obtain

[TABLE]

if we take $\varepsilon_{0}$ to be $J^{64}\varepsilon_{0}\leq 1$. It follows from (2.2), (4.49) and (4) that (4.52) holds. Note that (4) implies that there exists a positive constant $C_{7}$ independent of $J$ such that

[TABLE]

for $(r\omega,t)\in\mathcal{J}^{i}(\lambda;\omega)$. Therefore, by (1.21), (4.44), (4.54) and (4.58), we have

[TABLE]

Secondly, we will show

[TABLE]

for $i=1,2,\cdots,m$. Set $v=\partial_{0}u$ in (4.51), then we have

[TABLE]

By (1.1), (2.2), (2.17), (2.18), (2.19), (4.49) and (4.59), we have

[TABLE]

Hence, we have

[TABLE]

Set

[TABLE]

then (1.21), (4.45) and (4) imply the Cauchy problem of the ordinary differential equation;

[TABLE]

where

[TABLE]

Note that

[TABLE]

if we choose $\varepsilon_{0}$ to be $J^{3}\varepsilon_{0}^{\frac{1}{512}}<1$. Now we can show

[TABLE]

by using the following proposition.

Proposition 4.4

Let $w(t)$ be the solution of the ordinary differential equation;

[TABLE]

where $\alpha$ is a constant, $T_{0}$ and $T_{1}$ are positive constants and $q(t)$ is a continuous function in $[T_{0},T_{1})$. Assume

[TABLE]

Then,

[TABLE]

holds, as long as the right hand side of (4.70) is well-defined.

For the proof of Proposition 4.4, see the proof of Proposition 3.4 in [4].

By (4), we have $\displaystyle{q_{*}=\int_{t_{0}}^{T_{B}}|Q(s)|\ ds=O^{*}(\varepsilon^{\frac{513}{512}})<\infty}$ and

[TABLE]

for $0<\varepsilon<\varepsilon_{0}$, if we take $\varepsilon_{0}$ to be $(K_{2}B)^{512}\varepsilon_{0}<1$. Hence, it follows from (4.65), (4.66), (4), (4.70) and $HB<1$ that

[TABLE]

holds for $0<\varepsilon<\varepsilon_{0}$, if we tale $\varepsilon_{0}$ to be $\varepsilon_{0}\leq\{(1-HB)/(K_{2}B)\}^{512}$. Therefore, by (1.20), (4.58), (4.66) and (4), we obtain

[TABLE]

This implies (4.69). Moreover, by (2.2), (4.49) and (4.69), we have

[TABLE]

Now we show (4.60). Set $v=\Gamma_{p}u\ \ (p=3,4)$ in (4.51), then we have

[TABLE]

By (1.1), (2.2), (2.17), (2.18), (2.19), (4.49), (4.52) and (4.72), we have

[TABLE]

Hence, by setting

[TABLE]

we have

[TABLE]

if we take $\varepsilon_{0}$ to be $J\varepsilon_{0}\leq 1$. Thus, the Gronwall inequality implies

[TABLE]

Hence, by (4.43), (4) and (4), we have

[TABLE]

if we take $\varepsilon_{0}$ to be $J^{64}\varepsilon_{0}\leq 1$. Therefore, (4.38), (4.49) and (4) imply (4.60).

Finally, for any integer $h$ so that $2\leq h\leq k$, we show

[TABLE]

for $i=1,\cdots,m$, under the assumption

[TABLE]

Set $v=\Gamma^{a}u$ with $|a|\leq h$ in (4.51), then (1.1), (2.17), (2.19), (4.38), (4.39), (4.52), (4.72) and (4.80) imply

[TABLE]

and

[TABLE]

Thus, by setting

[TABLE]

and by (4.58), (4) and (4), we have

[TABLE]

The gronwall inequality implies

[TABLE]

Hence, by (4.58), we have

[TABLE]

It follows from (4.43) and (4) that (4.79) holds.