I have this beautiful Non-linear PDE $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$
Where C is a function of (x,t) It comes from the diffusion equation where D is concentration depending, and has the linear form $D=k \cdot C$ and is furthermore made dimensionless.
To solve this numerically I would like to first Linearize the PDE (if this is even possible), because i know how to solve linear PDE's, but how can I do this, is there any methods, articles or books which shows some procedures? or is linearization unnecessary and the PDE can be solved directly?