发表机构
Vanderbilt University（范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明二维空间中满足特定符号条件的三次拟线性波动方程存在全局解，并引入几何弱零条件，揭示其渐近行为异于线性波动方程。
AI 中文摘要
我们证明了一族二维空间中的标量拟线性波动方程，对于足够小的$C_c^\infty$初始数据，在未来和过去两个方向上都存在全局解。这些方程具有形如$u^2\partial^2u$的三次主部拟线性非线性项，并满足一个允许三次零条件失效的符号条件。为了解释该符号条件，我们推导了二维空间中具有三次主部非线性项的拟线性波动方程的几何约化系统，并引入了几何弱零条件的概念。我们证明了模型标量方程满足几何弱零条件当且仅当符号条件成立。在几何约化系统的层面上，符号条件阻止了相应的特征曲线在有限时间内相交。我们还从主定理中的全局解中恢复了几何约化系统。如果模型标量方程满足符号条件但三次零条件失效，且$u$是模型方程的非零全局解，我们证明，对于$r=|x|$，在$|r-t|\ll t$的区域中，$t^{\frac{1}{2}}\partial_r^2u$的$L^\infty$范数随着$t\to\infty$而趋于无穷。因此，其渐近行为不同于具有$C_c^\infty$数据的线性波动方程$\Box \psi=0$的解。
英文摘要
We prove global existence, both to the future and to the past, for a family of scalar quasilinear wave equations in two space dimensions for sufficiently small $C_c^\infty$ initial data. These equations have cubic leading quasilinear nonlinearities of the form $u^2\partial^2u$ and satisfy a sign condition that allows the cubic null condition to fail. To explain the sign condition, we derive the geometric reduced system for quasilinear wave equations with cubic leading nonlinearities in two space dimensions and introduce a notion of geometric weak null condition. We prove that the geometric weak null condition holds for the model scalar equation if and only if the sign condition holds. At the level of the geometric reduced system, the sign condition prevents the corresponding characteristic curves from intersecting in finite time. We also recover the geometric reduced system from the global solutions in our main theorem. If the model scalar equation satisfies the sign condition but fails the cubic null condition, and if $u$ is a nonzero global solution to the model equation, we show that, for $r=|x|$, the $L^\infty$ norm of $t^{\frac{1}{2}}\partial_r^2u$ in a region where $|r-t|\ll t$ tends to infinity as $t\to\infty$. Thus, its asymptotic behavior differs from that of a solution to the linear wave equation $\Box ψ=0$ with $C_c^\infty$ data.
Comments76 pages
