Instabilities Appearing in 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 detailed 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. 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.
