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：非线性偏微分方程出现在物理学的许多领域，本文研究了一类源自宇宙学研究的有效场论（EFT）中的典型方程。具体而言，我们关注$1+1$维中的方程 $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t)$。长期以来已知，当 $\alpha >0$ 时，该方程的解会在有限时间内发散。我们研究这种发散随参数 $\alpha>0$ 和 $\beta\ge0$ 变化的性质。与人们可能认为的相反，即使 $\beta$ 非常大，发散也不会消失（注意，由于我们考虑固定初始条件，$\alpha$ 和 $\beta$ 无法通过缩放消除）。但当 $\alpha$ 固定时，$\beta$ 增大将导致发散出现得更晚。我们发现存在两种类型的发散，并讨论了这两种发散随参数选择变化的转变。除非对相应方程进行修改，否则爆破不可避免。我们的结果可以推广到$3+1$维。