Global existence for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions
In this article is considered the Cauchy problem \[ \begin{aligned} \Bigl(\partial_t^2-\Delta_x+m_i^2\Bigr)u_i=F_i(u, \partial u, \partial\partial_x u),\quad i&=1, 2, \ldots, N,\quad {\text{ in}}\quad (0, \infty)\times \mathbb R^3,\\ u(0, x)=\epsilon f(x),\quad \partial_t u(0, x)&=\epsilon g(x)\quad {\text{ for}}\quad x\in \mathbb R^3, \end{aligned} \tag{1} \] where \(u=(u_j)_{1\leq j\leq N}\), \(m_i\geq 0\) for \(1\leq i\leq N\), \(\epsilon\) is a small and positive parameter, each component \(u_j\) of \(u\) is supposed to be a real-valued unknown function of \((t, x)\in (0, \infty)\times \mathbb R^3\), \(\Delta_x\) denotes the Laplacian in \(x\), \(\partial u=(\partial_a u_j)_{1\leq j\leq N, 0\leq a\leq 3}\), \(\partial_0={{\partial}\over {\partial t}}\), \(\partial_k={{\partial}\over {\partial x_k}}\), \(1\leq k\leq 3\), \(F_i\) is a real-valued smooth function of its arguments, \(1\leq i\leq N\). The author gives conditions for \(F=(F_i)_{1\leq i\leq N}\) for which for any \(f, g\in {\mathcal S}(\mathbb R^3; \mathbb R^N)\) there exists a positive constant \(\epsilon_0\) such that for every \(\epsilon\in (0, \epsilon_0]\) the Cauchy problem \((1)\) admits a unique global solution \(u\in {\mathcal C}^{\infty}([0, \infty)\times \mathbb R^3, \mathbb R^N)\), where \({\mathcal S}\) denotes the Schwartz class.
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