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This paper studies the Cauchy problem for systems of semi-linear waveequations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the nullconditions. We construct future global-in-time classical solutions witharbitrarily large initial energy. The choice of the large Cauchy initial datais inspired by Christodoulou's characteristic initial data in his work\cite{Ch-08} on formation of black-holes. The main innovation of the currentwork is that we discovered a relaxed energy ansatz which allows us to provedecay-in-time-estimate. Therefore, the new estimates can also be applied instudying the Cauchy problem for Einstein equations.

On classical global solutions of nonlinear wave equations with large data