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I have some problem to find a method to solve the following $PDE$: $$\partial_t ln[u(x,t)]=k^2\partial_{xx}u(x,t)$$ The equations resembles a common heat equation, but the logarithm of the function $u(x,t)$ seems to complicate the solution. Does someone have hints or answer? Many thanks.

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Does your PDE equivalent to $\partial_tu(x,t)=k^2u(x,t)\partial_{xx}u(x,t)$ ? – doraemonpaul Nov 12 '12 at 2:43
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If you try to write a solution as a complex Fourier Series, this should break down to a series expansion of $ax+b$. Of course, you will need two boundary conditions to fully solve the equation.

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This is nonlinear 2nd order PDE. I don't think writing the solution as a complex Fourier Series is easy. – doraemonpaul Jul 9 '12 at 8:33

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