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This dissertation describes the space of heteroclinic orbits for a class ofsemilinear parabolic equations, focusing primarily on the case where thenonlinearity is a second degree polynomial with variable coefficients. Alongthe way, a new and elementary proof of existence and uniqueness of solutions isgiven. Heteroclinic orbits are shown to be characterized by a particularfunctional being finite. A novel asymptotic-numeric matching scheme is used touncover delicate bifurcation behavior in the equilibria. The exact nature ofthis bifurcation behavior leads to a demonstration that the equilibria aredegenerate critical points in the sense of Morse. Finally, the space ofheteroclinic orbits is shown to have a cell complex structure, which is finitedimensional when the number of equilibria is finite.

Eternal solutions and heteroclinic orbits of a semilinear parabolic equation