Question

$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$ K and u are constants. If no,how to get a asymptotic solution?ie,linearize

Answer 1

We may assume WLOG that $K=1$. One family of solutions is $$h(x,t) = \frac{v(x)}{a+bt}$$ for arbitrary constants $a,b$, where $v(x)$ is a solution of the ordinary differential equation $$v v'' + (v')^2 + a \mu v = 0$$

Another family of solutions is $$h(x,t) = a \left(W\left(b e^{(x+ct) \mu c/a}\right)+1\right)$$ for arbitrary constants $a,b,c$, where $W$ is the Lambert W function.