This essentially self-contained paper continues the study of a
simple PDE in which an unorthodox sign in the spacial boundary
condition destroys the usual Minimum Principle. The long-term
behavior of solutions with time-parameter set $(0,\infty)$ is
established, and this clarifies in analytic terms the
characterization of non-negative solutions which had been obtained
previously by probabilistic methods. The paper then studies by
direct methods bounded 'ancient' solutions in which the
time-parameter set is $(-\infty,0)$. In the final section,
Martin-boundary theory is used to describe all non-negative
ancient solutions in the most interesting case. The relevant
Green kernel density behaves rather strangely, exhibiting two
types of behavior in relation to scaling of its arguments. The
Martin kernel density, a ratio of Green kernel densities,
behaves more sensibly. Doob $h$-transforms illuminate the
structure.
As a somewhat surprising consequence of our Martin-boundary analysis,
we find that non-negative solutions to our parabolic-looking equation
satisfy an elliptic-type Harnack principle.