GLOBAL CLASSICAL SOLUTIONS OF SEMILINEAR WAVE EQUATIONS ON $\mathbb{R}^{3}\times \mathbb{T}$ WITH CUBIC NONLINEARITIES*

doi:10.1007/s10473-024-0105-3

Abstract

Abstract: In this paper, we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space $\mathbb{R}^3\times\mathbb{T}$. The semilinear nonlinearity is assumed to be of the cubic form. The main ingredient here is the establishment of the $L^2$-$L^\infty$ decay estimates and the energy estimates for the linear problem, which are adapted to the wave equation on the product space. The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction, the scaling technique, and the combination of the decay estimates and the energy estimates.

Key words: semilinear wave equation, product space, decay estimate, energy estimate, global solution

CLC Number:

35L05

Fei Tao. GLOBAL CLASSICAL SOLUTIONS OF SEMILINEAR WAVE EQUATIONS ON $\mathbb{R}^{3}\times \mathbb{T}$ WITH CUBIC NONLINEARITIES^*[J].Acta mathematica scientia,Series B, 2024, 44(1): 115-128.

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GLOBAL CLASSICAL SOLUTIONS OF SEMILINEAR WAVE EQUATIONS ON $\mathbb{R}^{3}\times \mathbb{T}$ WITH CUBIC NONLINEARITIES^*

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