Eternal Solutions and Heteroclinic Orbits of a Semilinear Parabolic Equation
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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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2008-04-25T17:51:51Z
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Floer homology; semilinear parabolic equation; blow-up behavior; IMEX method; asymptotic series
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dissertation or thesis