Regarding some notation in writting Differential Equation.

Question

An initial value parabolic equation is of the form $$u_t+Au=f(t),~~t>0 $$ and $$u(0)=\phi.$$ For $\tau >0$, final value parabolic problem is of the form $$u_t+Au=f(t),~~0\leq t<\tau $$ and $$u(\tau)=\phi.$$ Now my question is that why do we write range of $t$ for initial value problem $t>0$ and for final valu problem $0\leq t<\tau?$

Answer 1

The initial value problem has a given $u$ for $t=0$ and the PDE is given for $t>0$.

The final value problem has a given $u$ for $t=\tau$ and the PDE is given for $t < \tau$.

So that problem model seems symmetric.

In both cases $t \ge 0$ is asked for. I suspect this is because one wants to use a certain solution technique, like Laplace transformations.