Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when $\alpha >0$. We study the nature of this divergence as a function of the parameters $\alpha>0 $ and $\beta\ge0$. The divergence does not disappear even when $\beta $ is very large contrary to what one might believe (note that since we consider fixed initial data, $\alpha$ and $\beta$ cannot be scaled away). But it will take longer to appear as $\beta$ increases when $\alpha$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions.
翻译：非线性部分方程式出现在物理的许多领域, 我们在这里研究一个典型的方程式, 人们从宇宙学研究的有效野战理论( EFT) 中发现一个典型方程式。 特别是, 我们感兴趣的是等式$\ part_ t ⁇ 2 u( x, t) =\ alpha( parte_ x u( x, t)) =\ pha( parte_ x( x, t)) ⁇ 2 ⁇ beta\ parte_ x% 2 u( x, t) $( x, t) 美元) 。 我们早就知道这个方程式的解决方案在一定时间里, $\ alpha > 0美元。 我们研究这种差异的性质是参数的函数 $\ alpha > 0 $ 和 $\ beta\ ge0 。 即使 $\ $\ beta $ 与人们可能相信的值非常大( 注意, 因为我们认为固定的初始数据不能缩放, $\ $\ $\\\\\\\\\ $\ $\ bebeata $ $ $ $ $ $ $ $ $) 。 但是它会显得更长化为美元。 但。 我们的变换成两个方程式的变形的参数是不可变形的。
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