Is the 1D Heat Equation with Dirichlet and Neumann Boundary Conditions on the Same Side Well Posed?

Question

If I take the 1D heat equation \begin{align} \frac{\partial u}{\partial t}=\kappa \frac{\partial^2u}{\partial x^2} \end{align} on some finite interval, say $0<x<1$, and then specify Dirichlet and Neumann boundary conditions on the righthand boundary, say \begin{align} u(1,t) = C, \frac{\partial u}{\partial x}\Bigr|_{x=1} = 0 \, \forall t \end{align} along with an initial value \begin{align} u(x,0) = f(x) \end{align} does this have a well posed solution? I would think not, since I would expect that there would need to conditions specified at the whole boundary (i.e. $x=0$ and $x=1$) for the solution to be well posed. However, that seems odd to me since there is a clear steady state solution for these boundary conditions, as $\partial u/\partial t = 0$ implies that the solution is just $u(x) = C$. Perhaps that's not strange at all, but my experience with PDEs is limited. Thanks!