Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions
The authors study the Cauchy problem of a quasi-linear system of two Klein-Gordon equations: \[ \square v+ M^2 v= F(v,\partial_t v,\partial_x v,\partial_t\partial_x v, \partial_x^2v),\;v|_{t= 0}= \varepsilon f,\;\partial_t v|_{t=0}=\varepsilon g. \] Here \(v= v(t,x_1,x_2)\in\mathbb{R}^2\), \(M: 2\times 2\) diagonal matrix with two positive diagonal elements \(m_1\), \(m_2\), and the degree of the real quadratic map \(F\) relatively to \((\partial_t\partial_x v,\partial^2_x v)\) is at most 1. They use coordinates \((T,X)\) defined by \(t+ 2B=T\cosh| X|\), \(x= TX\sinh| X|/| X|\), and derive the Cauchy problem: \[ (P_\kappa+ T^{-1} P^\Gamma)= T^{-1} Q(T, X,w,D_T w,T^{-1} D_x w), \] \(Q\): quadratic, with \(w|_{T= T_0}= w^0\), \(\partial_T w|_{T= T_0}= w^1\). Here \(v(t,x)= T\cosh(\kappa| X|))^{-1} w(T,X)\). Suppose that either ``\(m_1\neq 2m_2\) and \(m_2\neq 2m_1\)'' or ``\(m_1= 2m_2\) and the null condition of \(F\)'' is satisfied. They prove the theorem: There exists \(\sigma_0\in \mathbb{N}\) such that for any \(\sigma\in \mathbb{N}\); \(\sigma\geq \sigma_0\), one finds the following \(\varepsilon_0> 0\): For \(\forall\varepsilon\in (0,\varepsilon_0)\) and for any real valued \((w^0, w^1)\in H^\sigma\times H^{\sigma-1}\), supported inside \(\{| x|\leq \widetilde B\}\), satisfying \(\| w^0\|_{H_\sigma}+\| w^1\|_{H_{\sigma-1}}\leq A\varepsilon\), the equation in the derived problem has a unique global solution \(w\in C^0([T_0, \infty), H^\sigma)\cap C^1([T_0, \infty), H^{\sigma-1})\).
scientific article; zbMATH DE number 2105587