Global existence for a class of systems of nonlinear wave equations in three space dimensions
Consider a system of nonlinear wave equations \[ (\partial^2_t-c_i^2\Delta_x) u_i=F_i(u,\partial u,\partial_x\partial u)\quad\text{in } (0,\infty)\times \mathbb{R}^3 \] for \(i=1,\dots,m\), where \(F_i(i=1,\dots,m)\) are smooth functions of degree 2 near the origin of their arguments, and \(u=(u_1,\dots,u_m)\), while \(\partial u\) and \(\partial_x\partial u\) represent the first and second derivatives of \(u\), respectively. The author presents a new class of nonlinearity for which the global existence of small solutions is ensured. For example, global existence of small solutions for \[ \begin{cases}(\partial^2_t-c^2_1 \Delta_x)u_1=u_2 (\partial_tu_2)+\text{ arbitrary cubic terms},\\ (\partial^2_t-c^2_2 \Delta_x)u_2=u_1(\partial_tu_2)+(\partial_tu_1)u_2+\text{ arbitrary cubic terms}\end{cases} \] will be established, provided that \(c^2_1\neq c^2_2\).
scientific article; zbMATH DE number 2152522