Eternal Solutions and Heteroclinic Orbits of a Semilinear Parabolic Equation
نویسنده
چکیده
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. Michael Robinson entered the field of mathematics by a rather circuitous route. In high school, he learned computer programming as a hobby. It was there that his first exposure to partial differential equations occurred, when he wrote a solver for the invicid Navier-Stokes equations in the plane under the direction of Robert Ryder (who was then with Pratt & Whitney). Feeling that he ought to understand computer hardware at a deeper level, he enrolled at Rensselaer Polytechnic Insti-iii This project is dedicated to my wife, Donna, who urged me to do the obvious thing and continue my graduate studies. iv ACKNOWLEDGEMENTS I would like to thank all of the members of my thesis committee for their helpful and insightful discussions concerning this project. I would especially like to thank my advisor, Dr. John Hubbard for suggesting that I study ∂u ∂t = ∆u − u 2 + φ as a " summer project. " This problem has led into a surprising variety of interesting mathematics.
منابع مشابه
Construction of Eternal Solutions for a Semilinear Parabolic Equation
Abstract. Eternal solutions of parabolic equations (those which are defined for all time) are typically rather rare. For example, the heat equation has exactly one eternal solution – the trivial solution. While solutions to the heat equation exist for all forward time, they cannot be extended backwards in time. Nonlinearities exasperate the situation somewhat, in that solutions may form singula...
متن کاملA Cell Complex Structure for the Space of Heteroclines for a Semilinear Parabolic Equation
It is well known that for many semilinear parabolic equations there is a global attractor which has a cell complex structure with finite dimensional cells. Additionally, many semilinear parabolic equations have equilibria with finite dimensional unstable manifolds. In this article, these results are unified to show that for a specific parabolic equation on an unbounded domain, the space of hete...
متن کاملOn Connecting Orbits of Semilinear Parabolic Equations on S
It is well-known that any bounded orbit of semilinear parabolic equations of the form ut = uxx + f(u, ux), x ∈ S 1 = R/Z, t > 0, converges to steady states or rotating waves (non-constant solutions of the form U(x − ct)) under suitable conditions on f . Let S be the set of steady states and rotating waves (up to shift). Introducing new concepts — the clusters and the structure of S —, we clarif...
متن کاملSolving the inverse problem of determining an unknown control parameter in a semilinear parabolic equation
The inverse problem of identifying an unknown source control param- eter in a semilinear parabolic equation under an integral overdetermina- tion condition is considered. The series pattern solution of the proposed problem is obtained by using the weighted homotopy analysis method (WHAM). A description of the method for solving the problem and nding the unknown parameter is derived. Finally, tw...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008