Solving a non-linear diffusion equation

Question

I'm currently trying to solve the equation $$ \frac{\partial C}{\partial t}= \frac{\partial}{\partial x}\left(\frac{D}{C}\frac{\partial C}{\partial x}\right), $$ where D is a constant and $C \equiv C(x,t).$ The initial condition on $C$ is $C(x,0)=N_0 \delta(x)$, but further reasonable boundary conditions may be assumed true. So far I've tried solving either it using its Fourier transform or by treating it similarly to the heat equation (http://www.math.toronto.edu/courses/apm346h1/20129/L9.html), but my attempts have been unsucsessful. Any idea on how to find a solution? Thanks in advance for the help.