In this paper, the global well-posedness of semirelativistic equations with a power type nonlinearity on Euclidean spaces is studied. In two dimensional $H^s$ scaling subcritical case with $1\le s\le 2$, the local well-posedness follows from a Strichartz estimate. In higher dimensional $H^1$ scaling subcritical case, the local well-posedness for radial solutions follows from a weighted Strichartz estimate. Moreover, in three dimensional $H^1$ scaling critical case, the local well-posedness for radial solutions follows from a uniform bound of solutions which may be derived by the corresponding one dimensional problem. Local solutions may be extended by a priori estimates.

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在某些次临界和临界情况下非线性半相对论方程的整体适定性

本文研究了具有幂级数非线性的半相对论方程在欧式空间上的整体适定性。二维$H^s$ 扩展亚临界情况 $\mathrm{1个}\le s\le 2$，当地的适定性来自于Strichartz的估计。高维$H^{\mathrm{1个}}$在亚临界情况下，径向解的局部适定性来自加权的Strichartz估计。而且，在三维$H^{\mathrm{1个}}$在临界情况下，径向解的局部适定性来自解的统一边界，该边界可以由相应的一维问题得出。可以通过先验估计来扩展本地解决方案。