I am interested in finding the analytical solution of the following PDE, but I am not sure if it has one.
$$\frac{\partial{h}}{\partial{t}}=\gamma\nabla.(h\nabla h)$$ $$\implies\frac{\partial{h}}{\partial{t}}=\gamma\left[\nabla h.\nabla h + h\nabla^2 h\right]$$ Boundary conditions: $$\lim_{r \to 0} \left[2\pi r\gamma h \frac{\partial{h}}{\partial{r}}\right]=-a$$ $$[n.(\gamma h\nabla h)]_{r_n}=0$$
In above equations, $\gamma$ and $a$ are constants.
Domain describing $h(x,y,t), r, x, y, z$
I need the analytical solution for the above PDE in order to optimize the equation in terms of its time series for some fixed spatial coordinate.