I'd like to find analytical solutions of that kind of differential equations :

$$\partial_t c = \partial_x (D(c)\partial_x c) $$

with $D(c)$ a polynomial. The trivial cas $D(c)=a$ with $a$ a constant is the classical diffusion equation in 1d.

For polynomial such as : $D(c)= (a+bc)^n$ with $n$ an integer, the solution can be infered by a change of variable from the simple case $D(c)= C^n$.

In this article the author finds analytical solutions for $D(c)=D_0 (c/C_0)^n$ with $D_0 \; ,\; C_0$ some constants for a diffusion from a point. The method to find his result is to look for scaling functions like what I'm writing there.

I'd like to know if there are analytical solution for the case $D(c)=(c-a)(c-b)$ with $a\neq b$ some constants. I'm interested even in particular cases.

So I'd be happy if you have some references or some technics to tackle this problem, and it's even better if you have the result for a particular case :)

I'm not even sure the solution exists all the time so if you know articles or results about that thank you in advance.