I am trying to understand the behavior of the following linear PDE:

$$\partial_t u(x,t) = \partial_{xx} u(x,t) + f(x) u(x,t)$$

where I set $f(x)=\lambda e^{-x^2} -1$ and with:

IC : $u(x,0)= 1$ on $[-1,1]$ and $0$ elsewhere.

BC : Neumann $\partial_x u(\pm 1,t) = 0$

Numerically, it seems that there is a critical value $\lambda^* \approx 1.33$ such that: if $\lambda>\lambda*$ then $u(x,t)\to \infty$ and otherwise $u(x,t)\to 0$.

Can you explain me how to find this critical value ?

THANK YOU !!!