The partial null conditions and global smooth solutions of the nonlinear wave equations on \(\mathbb{R}^d \times \mathbb{T}\) with \(d = 2, 3\) (Q6065764)

In this paper, the authors investigate the Cauchy problem of the fully nonlinear wave equations, of the form \((\partial_t^2-\Delta_x-\partial_y^2) u=N(\partial u, \partial^2 u)\), on the product space \((x,y)\in\mathbb{R}^3 \times \mathbb{T}\) with quadratic nonlinearities and on \(\mathbb{R}^2 \times \mathbb{T}\) with cubic nonlinearities, respectively. When \(u\) is independent of \(y\), it is well-known that the nonlinearity is at the critical level for global existence with small data, and generally admits almost global existence. In addition, null condition is a sufficient condition to ensure global existence. For the problem in \(\mathbb{R}^n \times \mathbb{T}\), the authors introduce the partial null conditions, which are basically the null condition for \(N(\partial u, \partial^2 u)\) with \(u\) independent of the variable \(y\). The main results are the global existence under the partial null conditions, and the almost global existence, with lower bound of the lifespan \(T_\varepsilon\ge \exp(c\varepsilon^{-2/(n-1)})\), for the small initial data of size \(\varepsilon\) with sufficient decay and smoothness. The proof relies on the Fourier mode decomposition of the solutions with respect to the periodic variable, the efficient combinations of time-decay estimates for the solutions to the linear wave equations and the linear Klein-Gordon equations, and the global weighted energy estimates.