# Does this PDE has a general solution? [closed]

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

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## closed as too localized by Ricky Demer, Qiaochu Yuan, Andy Putman, Michael Renardy, Pietro MajerNov 19 '12 at 17:38

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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.

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"We may assume WLOG that $\: K=1 \:$" $\;\;$ except when $\: K=0 \:$. $\;\;\;\;$ – Ricky Demer Nov 18 '12 at 5:44
OK, but the case $K=0$ is left as an exercise. – Robert Israel Nov 18 '12 at 8:09