is it possible to have $u(1,t) = 10$ and $u(x,0) = x^2$ for $x \in [0,1]$? (heat equation) 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.

The initial conditions should really be stated inside of the domain, not including the boundary: $u(x, 0) = x^2 $ for $x\in (0, 1)$.  That's because the solution is really defined on an open set, and the boundary conditions are a statement about the limits of $u(x, t)$ as $x$ approaches a boundary point.  
But it's common to gloss over all this in textbooks. One point does not make a difference anyway (unless there is a point source / Dirac delta in there). 
So, the boundary conditions make sense and can be satisfied, with the understanding that there will not exist a limit of $u(x, t)$ as $(x, t) \to (1, 0)$. At very small scale, the behavior of $u(x, t)$ near $(1, 0)$ will be complicated and not really consistent with physics (transfer of energy at unbounded rate). But ignoring that tiny bit, the rest of the function will be reasonable. 
