AI 中文总结
该研究确定了带强阻尼的三维半线性波动方程的临界指数为7/3，解决了2014年的开放问题，通过相关分析得到了不同幂次下解的全局存在与爆破结果。
AI 中文摘要
在本手稿中，我们确定了带强阻尼的三维半线性波动方程的临界指数，从而解决了一个2014年提出的开放问题。幂非线性项|u|^p的阈值为p=p_crit=7/3。当p>7/3时，足够小的初值会产生时间上全局存在的解；而当1<p≤7/3时，存在任意小的光滑紧支初值，其解会在有限时间内爆破。全速度基本解的正性结合降维公式，得到了正半轴核，替代了强阻尼波所不具备的有限传播性质。这引出了一个非线性下界抛物迭代，无需径向对称性或对初值的逐点符号假设。在临界幂次下，对移动壳的精细切片论证将边界对数增益转化为爆破所需的增长。
英文摘要
In this manuscript, we determine the critical exponent for the three-dimensional semilinear wave equation with strong damping and thereby resolve an open problem posed in 2014. The threshold for the power nonlinearity $|u|^p$ is \begin{align*} p=p_{\mathrm{crit}}=\frac{7}{3}. \end{align*} Sufficiently small data generate global in-time solutions for $p>\frac{7}{3}$, whereas there exist arbitrarily small smooth compactly supported data whose solutions blow up in finite time for $1<p\leqslant\frac{7}{3}$. Positivity of the full velocity fundamental solution, combined with a dimension-descent formula, yields a positive half-line kernel and replaces the finite propagation property unavailable for strongly damped waves. This leads to a nonlinear lower-bound parabolic iteration without radial symmetry or pointwise sign assumptions on the initial data. At the critical power, a refined slicing argument on moving shells converts the borderline logarithmic gain into the growth required for blow-up.
