Global existence for systems of wave equations with nonresonant nonlinearities and null forms
The paper deals with the system of nonlinear wave equations of the form \[ \partial^2_t u_i- c^2_i\Delta_x u_i= F_i(u, \partial u,\nabla_x\partial u)\text{ in }(0,\infty)\times \mathbb{R}^3\;(1\leq i\leq m) \] with initial data \[ u_i(0,x)=\varepsilon f_i(x),\;\partial_ tu_i(0,x)=\varepsilon g_i(x),\quad x\in \mathbb{R}^3\;(1\leq i\leq m), \] where \(c_i\) is a given positive constant \((1\leq i\leq m)\) and \(F(u,v,w)= (F_j(u, v,w))^m_{j=1}\) is a function of \((u, v,w)= \mathbb{R}^m\times \mathbb{R}^{4m}\times \mathbb{R}^{12m}\) satisfying \(F(u,v,w)= 0\) \((|u|^2+ |v|^2+ |w|^2)\) near \((u,v,w)= (0,0,0)\). The author proves the global existence of small amplitude solutions for some nonresonant nonlinearities.
scientific article; zbMATH DE number 2136642
scientific article