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In this paper, we investigate the fully nonlinear wave equations on theproduct space $\mathbb{R}^3\times\mathbb{T}$ with quadratic nonlinearities andon $\mathbb{R}^2\times\mathbb{T}$ with cubic nonlinearities, respectively. Itis shown that for the small initial data satisfying some space-decay rates atinfinity, these nonlinear equations admit global smooth solutions when thecorresponding partial null conditions hold and while have almost global smoothsolutions when the partial null conditions are violated. Our proof relies onthe Fourier mode decomposition of the solutions with respect to the periodicdirection, the efficient combinations of time-decay estimates for the solutionsto the linear wave equations and the linear Klein-Gordon equations, and theglobal weighted energy estimates. In addition, an interesting auxiliary energyis introduced. As a byproduct, our results can be applied to the 4Dirrotational compressible Euler equations of polytropic gases or Chaplygingases on $\mathbb{R}^3\times\mathbb{T}$, the 3D relativistic membrane equationand the 3D nonlinear membrane equation on $\mathbb{R}^2\times\mathbb{T}$.

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