October  2004, 10(4): 941-964. doi: 10.3934/dcds.2004.10.941

Evans function and blow-up methods in critical eigenvalue problems

1. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210

2. 

Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received  November 2002 Revised  September 2003 Published  March 2004

Contact defects are one of several types of defects that arise generically in oscillatory media modelled by reaction-diffusion systems. An interesting property of these defects is that the asymptotic spatial wavenumber is approached only with algebraic order O$(1/x)$ (the associated phase diverges logarithmically). The essential spectrum of the PDE linearization about a contact defect always has a branch point at the origin. We show that the Evans function can be extended across this branch point and discuss the smoothness properties of the extension. The construction utilizes blow-up techniques and is quite general in nature. We also comment on known relations between roots of the Evans function and the temporal asymptotics of Green's functions, and discuss applications to algebraically decaying solitons.
Citation: Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941
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