$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$ K and u are constants. If no,how to get a asymptotic solution?ie,linearize
closed as too localized by Ricky Demer, Qiaochu Yuan, Andy Putman, Michael Renardy, Pietro Majer Nov 19 '12 at 17:38This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 


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. 

