Finite element method for quasilinear parabolic PDE

I have a quasilinear parabolic PDE like this: $H_t=(a-bH^{1/3})H_{xx}+(c+1/3bH^{-2/3}+bH^{1/3})H_x$ with standard homogeneous BC and IC. What methods can you suggest to solve it numerically? Is it possible to solve it with simple FEM directly? I tried mixed FEM, but i can not implement it due to 1/3 degrees. I found another methods like $H^1$ Galerkin method and collocation FEM that can be used in order to solve it. Do you have some ideas?