Fisher's equation

Remarks

The constants $u \equiv 0$ and $u \equiv 1$ are solutions: both sides vanish. The state $0$ is unstable and the state $1$ is stable for the reaction $u' = r u(1 - u)$. A travelling wave is a solution $u(x, t) = U(x - ct)$ with $U(-\infty) = 1$ and $U(+\infty) = 0$. Such a wave with $0 \le U \le 1$ exists if and only if $c \ge 2\sqrt{rD}$. A non-example is the heat equation $u_t = D u_{xx}$: it has no reaction term, so it is not Fisher's equation for any $r > 0$. The rescaling $x \mapsto x\sqrt{r/D}$, $t \mapsto r t$ reduces the equation to $u_t = u_{xx} + u(1 - u)$.