Literature on the PDE $\partial_t f + \partial_x(f(1-f))=0$

Question

I am looking for literature on this variant of the transport PDE:

$$\partial_t f + \partial_x(f(1-f))=0.$$

Do you have any suggestions?

Answer 1

This is a special case of a conservation law that can be (and has been) written in divergence form $$ \partial_t f + \partial_x F(f) = 0$$ where $F(x) = x(1-x)$. You can find some information about this class of equations in e.g. Evans' Partial Differential Equations, Chapter 3.4. IMO, a softer introduction can be found in Alinhac's book Hyperbolic Partial Differential Equations, Chapter 4. One small point, the theory at some points assumes $F$ is convex, or even uniformly convex, which we can achieve by setting $u = -f$ so that we have the equation $$ u_t + G(u)_x = 0,\quad G(x) = x(x+1)$$