Boundary conditions for second order PDE

Question

For a second order PDE, for example heat conduction equation $\frac{\partial T}{\partial t} = \frac{\alpha}{C_p} \nabla^2 T$, is it possible to determine the steady-state (or even transient) solution with two Dirichlet conditions? I have two different questions regarding this

1. From my understanding, the solution is non unique for all equal valued ratios of $\alpha$ and $C_p$. so two Dirichlet conditions say nothing about how fast the disturbance propagates with a temporal change of one boundary condition. So only the knowledge of $T$ and $\nabla T$ together can fix the solution curve for specific values of $\alpha$ and $C_p$ instead of the ratio.

2. Integrating the 1-D second order (steady-state) equation gives $T=C_1x+C_2$ where $C_1 = \frac{\partial T}{\partial x}$. So, two Dirichlet conditions are two values of $C_2$ and therefore still don't give us the value of $C_1$ which is required to fix the solution curve. So in this case, how is it possible to assume we know the solution with two just Dirichlet conditions?

Answer 1

You already have the steady state solution well defined.

If

$$T(x) = C_1x + C_2$$

and

$$T(x=0) = T_1, T(x=l) = T_2,$$

then

$$T(x) = \frac{T_2 - T_1}{l}x + T_1, 0 < x < l.$$

Answer 2

I'm assuming by Dirichlet boundary conditions, you mean that the heat at the endpoints of the rod are known and fixed for all time?

In this case, at steady state you have that $$0 = \frac{\alpha}{C_p} \Delta T.$$ This clearly does not depend on the constants, which divide out to give Laplace's equation $$0 = \Delta T$$ which is well known to have unique solutions given Dirichlet boundary conditions on a compact domain (to see this, suppose you had two solutions $T_1(x)$ and $T_2(x)$, consider $T_1 - T_2$, and invoke the maximum principle.) In the 1D setting, this solution is a linear function interpolating your boundary conditions.