Is there a name for the type of boundary condition where the initial boundary values are known but are not held constant over time?

Question

I'm exploring the heat equation to model a particular 1D scenario, and I understood the Dirichlet and Neumann boundary conditions, but neither are sufficient for my scenario. Assuming a rod of length L, I want the boundaries to have a particular initial value ($U(0,0) = 400$, $U(L,0) = 300$), but the temperatures at the boundaries do not need to be constant across time ($U(0,0) ≠ U(0,t)$, $U(L,0) \ne U(L,t))$. Heat does flow in and out of the boundary, but only towards the rod, not the air.

Now, my question is, is there any sort of name for this type of boundary condition, where the initial boundary values are known, and are not held constant over time?

I hope the explanation of my scenario was clear. Please drop a comment in case you need clarification on some point.

Answer 1

If there are boundary conditions $$ \phi_0(t) \equiv U(0,t),\\ \phi_1(t) \equiv U(L,t), $$ and the initial condition $$ f(x) \equiv U(x,0), $$ then you have what is called time-dependent boundary conditions, see e.g. these lecture notes. The problem statement in your question represents a special case, where only the values of $f(x)$ at two points are specified. I don't know of a more specific name for this case (and I wonder if the problem is actually underdetermined, but that is probably a question for Math.SE).

Answer 2

Well, after consulting my professor, it seems it was a Neumann boundary condition with zero flux at both boundaries ($\phi(0,t) = \phi(L,t) = 0$)