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?