  This dissertation describes the space of heteroclinic orbits for a class of
semilinear parabolic equations, focusing primarily on the case where the
nonlinearity is a second degree polynomial with variable coefficients. Along
the way, a new and elementary proof of existence and uniqueness of solutions is
given. Heteroclinic orbits are shown to be characterized by a particular
functional being finite. A novel asymptotic-numeric matching scheme is used to
uncover delicate bifurcation behavior in the equilibria. The exact nature of
this bifurcation behavior leads to a demonstration that the equilibria are
degenerate critical points in the sense of Morse. Finally, the space of
heteroclinic orbits is shown to have a cell complex structure, which is finite
dimensional when the number of equilibria is finite.
