Eternal solutions and heteroclinic orbits of a semilinear parabolic equation

Details Eternal solutions and heteroclinic orbits of a semilinear parabolic equation Eternal solutions and heteroclinic orbits of a semilinear parabolic equation

2008-04-30

arxiv.org/abs/0804.4883v1

Mathematics

Analysis of PDEs

170 pages, many figures

Scientific paper

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.

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