In [T.C. Sideris, Comm. Part. Diff. Eqs., 8:1291–1323, 1983], the author proves the solution of the nonlinear wave equations breaks down in finite time, if the initial data is radially symmetric and arbitrarily small. The present article is devoted to the study of the lower bound of blow-up rate of blow-up solution and the global solution to a class of nonlinear wave equations in $\mathbb{R}^d , d \gt 3$. We first recall some useful lemmas in Besov spaces. Next, the local well-posedness of Equation (1.1) is obtained in $\dot{B}^{\frac{d}{2} - \frac{1}{2}}_{2,1} \cap \dot{B}^{\frac{d}{2}}_{2,1}$, and a lower bound of blow-up rate of blow-up solution in the space is established. Finally, by construction of the space $\mathscr{X}_R (M)$, thanks to the contraction mapping argument, we derive the global solution for the Cauchy problem of Equation (1.1) if the initial datum is sufficiently small.

中文翻译：

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在$ \ mathbb {R} ^ d $中的非线性波动方程在Besov空间中的解的整体存在

在[TC Sideris，Comm。部分。差 等式，8：1291–1323，1983]，作者证明了如果初始数据是径向对称且任意小的，则非线性波动方程的解将在有限时间内分解。本文致力于研究$ \ mathbb {R} ^ d，d \ gt 3 $中一类非线性波动方程的爆破解的爆破率下界和整体解。我们首先回顾一下Besov空间中的一些有用引理。接下来，在$ \ dot {B} ^ {\ frac {d} {2}-\ frac {1} {2}} _ {2,1} \ cap \中获得方程式（1.1）的局部适定性点{B} ^ {\ frac {d} {2}} _ {2,1} $，并确定空间中爆炸解决方案的爆炸率下限。最后，通过构造空间$ \ mathscr {X} _R（M）$，得益于压缩映射参数，我们得出了方程Cauchy问题的整体解（1。