Singular perturbation and bifurcation of transition layer solutions in inhomogeneous media

Chaoqun Huang, Purdue University

Abstract

We study the singularly perturbed reaction diffusion equation in degenerate spatially inhomogeneous media. We introduce bifurcation parameter into the spatial inhomogeneity to study the similarities and the differences of bifurcation of transition layers and bifurcation of the zeros of the spatial inhomogeneity. Liapunov-Schmidt reduction technique is the main tool to the construction of equilibrium layers as a perturbation of some good approximate solution in the direction of the principal eigenfunction. Reduced bifurcation equation is the pivotal equation to prove the existence of bifurcation of transition layers. The derivation of the reduced bifurcation relies on careful study of a singularly perturbed linearized eigenvalue problem. Even though, it is found that the form of the reduced bifurcation equation inherits mostly from the form of the spatial inhomogeneity, there are subtle differences between the two. We have found explicit examples where bifurcation of transition layers and that of zeros of the spatial inhomogeneity are different in terms of imperfectness. For the transition layers problem, there are at least two natural ways to introduce imperfectness. One way is through adding higher-order variations to spatial inhomogeneity. Imperfectness comes into higher dimensional reaction diffusion problem naturally even if spatial inhomogeneity is radially symmetric, perfect pitchfork normal form.

Degree

Ph.D.

Advisors

Yip, Purdue University.

Subject Area

Applied Mathematics

COinS