Instabilities Appearing in Cosmological Effective Field theories: When and How?

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.