$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$ K and u are constants. If no,how to get a asymptotic solution?ie,linearize
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3$\begingroup$ mathoverflow.net/questions/25431/… $\endgroup$– Steve HuntsmanCommented Nov 18, 2012 at 4:59
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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.
answered Nov 18, 2012 at 5:19
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$\begingroup$ "We may assume WLOG that $\: K=1 \:$" $\;\;$ except when $\: K=0 \:$. $\;\;\;\;$ $\endgroup$– user5810Commented Nov 18, 2012 at 5:44
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2$\begingroup$ OK, but the case $K=0$ is left as an exercise. $\endgroup$ Commented Nov 18, 2012 at 8:09