Is there a general criterion in judging the rationality (or appropriateness) of the boundary conditions? When we consider a partial differential equation, we need to give some boundary condition such that this question is reasonable. What criterion can we use? e.g., heat equation $\partial_t u(t,x) = \Delta_x u(t,x)$ we take Laplace transfrom, then we get $\tau u(\tau,x) - u(0,x) = \Delta u(\tau,x)$. It seems that a boundary condition in the form of initial condtion $u(0,x) = f(x)$ is more reasonable. But is there a general criterion in judging the rationality (or appropriteness) of the boundary conditions?

The paper of Kurt Friedrichs, Symmetric Positive Linear Differential Equations, Communications in Pure and Applied Mathematics, vol 11, p333-458 (1958), discusses this question.
