If you have a diffusion equation $\partial_t f(x,t) = \partial_x^2 f(x,t) $, where $(x,t) \in [0,a] \times \mathbb{R}$ and then you say $\lim_{t \rightarrow -\infty} f(x,t) = y_0(x)$, how do you call such a boundary condition? My feeling is that this completely determines the solution, is this correct?
If anything is unclear about my question, please let me know.
I have the impression that you have to give boundary conditions in $x=0$ and $x=a$ in order to give a meaning to $\partial _{x}^{2}$. Then its spectrum becomes discrete \begin{equation*} \partial _{x}^{2}=\sum_{n}\lambda _{n}|u_{n}><u_{n}| \end{equation*} Usually the $\lambda _{n}$ are non-positive. Now \begin{equation*} f(x,t)=<x|f(t)>=\exp [\partial _{x}^{2}t]f(x,0)=\sum_{n}\exp [\lambda _{n}t]|u_{n}><u_{n}|f(0)>\overset{t\rightarrow -\infty }{\rightarrow } y_{0}(x) \end{equation*} Then the limit as $t\rightarrow -\infty $ will only exist for those $n$ for which $\exp [\lambda _{n}t]$ has a limit. Thus all other $<u_{n}|f(0)>$ must vanish. For positive $\lambda _{n}$ the corresponding terms vanish in the limit and hence the associated $<u_{n}|f(0)>$ are undetermined. Thus, for $% y_{0}\neq 0$ only those $n$ for which $\lambda _{n}=0$ contribute.
By the way, your condition is usually refered to as an asymptotic condition rather than a boundary condition.