If Initial conditions are 0 at both the endpoints of the pde, does that mean that for all time that the endpoints will be 0? Like in the graph, the initial graph is the blue and the solution (if it is right) is in red. Notice how at 0 and 1 the updated solution is no longer 0.
Initial conditions are just that: initial (telling what happens at the starting time, often taken to be $0$). Boundary conditions tell what happens at the boundary of the spatial region, e.g. $x = 0$ and $x = L$. The boundary conditions don't have to match the initial conditions (though that means the solution will be discontinuous at the corners), and they don't have to be constant in time.