I was looking over a problem that asks for the physical interpretation of a partial differential equation ($u_t = u_{xx}$ for $x \, \in \, (0,1)$, $t \geq 0$).
But the problem gives a boundary condition:
$$u(1,t) = 10, \hspace{0.35cm} \text{ for $t > 0$},$$
and the initial condition:
$$u(x,0) = x^2, \hspace{0.35cm} \text{ for $x \, \in \, [0,1{\color{red}]}$} $$
Am I mistaken in believing that this initial condition should have been stated as:
$$u(x,0) = x^2, \hspace{0.35cm} \text{ for $x \, \in \, [0,1{\color{red})}$} $$
instead? Since the boundary condition at $x=1$ requires $u(1,0) = 10$, or is this fine because the boundary condition is for $t > 0$ (which doesn't include $t = 0$)?
I doubt it's of importance to the concepts at hand, I was just curious.