How to get from heat equation with final condition to one with initial condition?

Question

How do I get from the heat equation with end condition $$\frac{d}{dt}u(x,t) + \Delta u(x,t) = f(x,t)$$ $$u(x,T) = u_0(x)$$ where $t \in (0,T)$ and $x \in \Omega$, to a normal heat equation with initial condition?

if I substitute $s= T-t$, I dont know how to treat the time derivative.

Answer 1

If you pose $\tau=T-t$ you have: $$ \frac{\partial}{\partial \tau}=\frac{\partial}{\partial (T-t)}=-\frac{\partial}{\partial t}, $$ then everything should flow.