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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Layers and spikes in non-homogeneous bistable reaction-diffusion equations


Authors: Shangbing Ai, Xinfu Chen and Stuart P. Hastings
Journal: Trans. Amer. Math. Soc. 358 (2006), 3169-3206
MSC (2000): Primary 35K57
Published electronically: February 20, 2006
MathSciNet review: 2216263
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Abstract: We study $ \varepsilon^2\ddot{u}=f(u,x)=A\, u\, (1-u)\,(\phi-u)$, where $ A=A(u,x)>0$, $ \phi=\phi(x)\in(0,1)$, and $ \varepsilon>0$ is sufficiently small, on an interval $ [0,L]$ with boundary conditions $ \dot{u}=0$ at $ x=0,L$. All solutions with an $ \varepsilon$ independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to $ f=A(u,x)\; (u-\phi_-)\,(u-\phi)\,(u-\phi_+)$ with $ \phi_-(x)<\phi(x)<\phi_+(x)$ and also to an infinite interval.


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Additional Information

Shangbing Ai
Affiliation: Department of Mathematical Sciences, University of Alabama at Huntsville, Huntsville, Alabama 35899
Email: ais@email.uah.edu

Xinfu Chen
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: xinfu@pitt.edu

Stuart P. Hastings
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: sph@pitt.edu

DOI: https://doi.org/10.1090/S0002-9947-06-03834-7
Received by editor(s): April 17, 2002
Received by editor(s) in revised form: September 1, 2004
Published electronically: February 20, 2006
Additional Notes: The second author thanks the National Science Foundation Grant DMS–0203991 for their support.
Article copyright: © Copyright 2006 American Mathematical Society