Construction of periodic solutions of nonlinear wave equations in higher dimension
The author deals with the existence of space and time periodic solutions of nonlinear wave equations \(u_{tt}- \Delta u+ F' (u)=0\). In particular, for the periodic wave equation in dimension \(d\) \[ u_{tt}- \Delta u+ \rho u+ u^3=0 \tag{1} \] the following proposition is demonstrated: Let \(\rho\in \mathbb{R}\) satisfy a condition of the form \[ \Biggl|\sum^r_{j=0} a_j \rho^j \Biggr|> \Bigl(\sum |a_j|\Bigr)^{- c(r)} \qquad \text{for all } \{a_j\}\in \mathbb{Z}^{r+1} \setminus \{0\}. \] Fix \(m_0\in \mathbb{Z}^d \setminus \{0\}\). Then there is a Cantor set \(C\) of positive measure in an interval \([0, \delta ]\) such that for \(p_0\in C\) there exists a solution of (1) of the form \(u= p_0 \cos (\langle m_0, x\rangle+ \lambda t)+ O(p^3_0)\), where \(\lambda^2= |m_0 |^2+ \rho+ {3\over 4} p^2_0+ O(p^4_0)\).
scientific article; zbMATH DE number 800441