具强阻尼的二维半线性波动方程的临界指数

中文摘要 AI 辅助

本文确定了具强阻尼的二维半线性波动方程的临界指数。D'Abbicco（arXiv，2026）的最新结果表明，当p>10/3时，该方程存在全局时间的小初值解；而本文在初速度满足平均符号条件的情况下，证明了当2<p≤10/3时解会在有限时间内爆破。结合先前已知的1<p≤3时的爆破结果，幂非线性项|u|^p的阈值为p=p_crit=10/3。本文还证明了二维速度基本解的正性，并构造了一个正的平均核，这使我们能在抛物区域建立非线性下界迭代。

英文摘要

In this manuscript, we determine the critical exponent for the two-dimensional semilinear wave equation with strong damping. A recent result in D'Abbicco (arXiv, 2026) gives global in-time small data solutions for $p>\frac{10}{3}$, whereas we in the paper prove finite-time blow-up for $2<p\leqslant\frac{10}{3}$ under an averaged sign condition on the initial velocity. Together with the previously known blow-up result for $1<p\leqslant3$, the threshold for the power nonlinearity $|u|^p$ is \begin{align*} p=p_{\mathrm{crit}}=\frac{10}{3}. \end{align*} We prove positivity of the full two-dimensional velocity fundamental solution and construct a positive averaged kernel, which allows us to establish a nonlinear lower-bound iteration in a parabolic region.

发表机构

• Guangzhou University（广州大学）

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