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I am solving a problem which goes like this:

Solve the diffusion equation:$$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$$ for $t>0$ and $x \in > \mathbb{R}$, with initial condition $u(x,0)=e^{-x}$. Interpret the result in terms of heat conduction.

I have solved the equation and the solution is simply $u(x,t)=e^{t-x}$, but I haven't managed to interpret it in terms of heat conduction. I understand that it corresponds to the heat equation in an infinite bar with conduction coefficient $\kappa=1$, but, how it is possible that the solution be, not only divergent but also that it increases with time?

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It is a result of the boundary condition. If for $t = 0$, the function has a positive second derivative, (it is concave upwards as $e^{-x}$) it will increases with time.

It could be a long rod, while one of the ends is being heated (since t < 0) at a rate greater than the flow of heat by conduction through it.

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