Introduction and main result

We consider the Cauchy problem for systems of semilinear wave equations with different propagation speeds in three space dimensions of the form

$$\square_{c_i}u_i=F_i(u,\partial_t u), \quad (x,t)\in \mathbb{R}^3\times [0,\infty),\, i=1,2,$$ (1.1)

$$u_i(x,0)=\varepsilon \varphi_i(x), \hspace{3mm} \partial_t u_i(x,0)=\varepsilon \psi_i(x), \quad x\in \mathbb{R}^3,\, i=1,2,$$ (1.2)

where $\square_{c}=\partial_t^2-c^2\Delta$, $c_1$, $c_2$, $\varepsilon$ are positive constants, $c_1\ne c_2$, and $u=(u_1,u_2)$ is an $\mathbb{R}^2$-valued unknown function of $(x,t)$. We assume that the nonlinear functions $F_1$ and $F_2$ are quadratic with respect to $(u,\partial_t u)$, and study the small data global existence and blowup for (1.1). Here, we say that the small data global existence holds for (1.1) if for any $\varphi_i$, $\psi_i\in C_0^{\infty}(\mathbb{R}^3)$ $(i=1,2)$ there exists a constant $\varepsilon_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0]$ the Cauchy problem (1.1)-(1.2) admits a unique global classical solution $u\in C^{\infty}(\mathbb{R}^3\times [0,\infty),\mathbb{R}^2)$. Moreover, we say that the small data blowup occurs if the small data global existence does not hold. In the present paper, we do not consider the case where the nonlinear terms $F_i$ depend only on $u$ (for that case, see Kubo and Ohta [10]), and we put

$$F_i(u,\partial_t u)=\sum_{j,k=1,2} (A_{i}^{j,k}u_j\partial_t u_k+B_{i}^{j,k}\partial_t u_j\partial_t u_k),$$ (1.3)

where $A_{i}^{j,k}$, $B_{i}^{j,k}\in \mathbb{R}$, $i=1$, $2$. In what follows, we always assume that

$$A_{i}^{i,i}=B_{i}^{i,i}=0,\quad i=1,2,$$ (1.4)

because it is proved by F. John [4] that the small data blowup occurs for the single equations $\square u=u\partial_t u$ and $\square u=(\partial_t u)^2$ in three space dimensions (see Klainerman [8] and Christodoulou [2] for the small data global existence when $c_1=c_2$).

For the case $c_1\ne c_2$ and (1.4), the small data global existence for (1.1) has been studied by many authors (see, e.g., [1,3,5,6,7,9,11,12,13]). Yokoyama [13] proved that the small data global existence holds for (1.1) with (1.3) if $c_1\ne c_2$ and $A_{i}^{j,k}=B_{i}^{i,i}=0$ for $i,j,k=1,2$. For the case where both $F_1$ and $F_2$ can be written in the divergent form

$$F_i=\partial_t \left(\sum_{j,k=1,2}D_{i}^{j,k}u_ju_k\right), \quad D_{i}^{j,k}\in \mathbb{R},~ i=1,2,$$

it is proved in [5] that the small data global existence holds for (1.1) if $c_1\ne c_2$ and $D_{i}^{i,i}=0$ for $i=1,2$. Moreover, Katayama [7] proved that the small data global existence holds for (1.1) with (1.3) if $c_1\ne c_2$ and $A_{i}^{j,j}=B_{i}^{j,j}=0$ for $i,j=1,2$.

However, to our knowledge, no results on the small data blowup have been obtained for (1.1) with (1.3) when $c_1\ne c_2$ and (1.4). The purpose in the present paper is to show that the condition (1.4) is not sufficient to prove the small data global existence for (1.1) with (1.3) when $c_1\ne c_2$. More precisely, we consider

$$\left\{\begin{array}{ll} \square_{c_1}u_1=u_2\partial_t u_1, &\qu... ..._2(x,0)=0,~ \partial_t u_2(x,0)=0, &\quad x\in \mathbb{R}^3. \end{array}\right.$$ (1.5)

The main result in the present paper is as follows.

Theorem 1.1   Let $0<c_1<c_2$, $\varepsilon\in (0,1]$, $\psi_1(\vert x\vert)\in C_0^{\infty}(\mathbb{R}^3)$, and we assume that there exists a constant $\delta>0$ such that

$$\psi_1(r)>0 \hspace{2mm} r\in [0,\delta),\quad \psi_1(r)=0 \hspace{2mm} r\in [\delta,\infty).$$ (1.6)

Then, the classical solution $(u_1,u_2)$ of $(\ref{b-sys})$ blows up in a finite time $T^*(\varepsilon)$. Moreover, there exists a positive constant $C^*$, which is independent of $\varepsilon$, such that

$$T^*(\varepsilon)\le \exp(C^*\varepsilon^{-2}),\quad \varepsilon\in (0,1].$$

In the next section, we will give the proof of Theorem 1.1.