I'm interested in solving the partial differential equation : $$\frac{\partial f(t,x)}{\partial t}+\frac{\partial^2 f(t,x)}{\partial x^2}=0$$ and $f(0,x)=f_{in}(x)$ with $(t,x)\in\mathbb{R}^+\times\mathbb{R}^+$ (or smaller set)
which is like a time-reverse heat equation. I know the solution might not be defined for all $t$ but is there an integral formula to solve this equation similar to the heat equation ?
Trying to use the heat kernel, I found that the function $$f(t,x)=\frac{C}{\sqrt{t}}\exp\left(\frac{x^2}{4t}\right)$$ is a solution (but not defined in $t=0$) but I couldn't find a general solution for a given $f_{in}$.
You already run into problems because you cannot impose the initial data at $t=0$. In fact if you try to solve it you end up with solutions that are unbounded in arbitrarily small intervals ! That's why we say that the problem as stated is ill-posed.