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Concentrating patterns of reaction-diffusion systems: A variational approach


Authors: Yanheng Ding and Tian Xu
Journal: Trans. Amer. Math. Soc. 369 (2017), 97-138
MSC (2010): Primary 35A15, 35K57, 49J35
DOI: https://doi.org/10.1090/tran/6626
Published electronically: March 1, 2016
MathSciNet review: 3557769
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Abstract: Our purpose is to motivate an analytic characterization aimed at predicting patterns for general reaction-diffusion systems, depending on the spatial distribution involved in the reaction terms. It is shown that there must be a pattern concentrating around the local minimum of the chemical potential distribution for small diffusion coefficients. A multiple concentrating result is also established to illustrate the mechanisms leading to emergent spatial patterns. The results of this paper were proved by using a general variational technique. This enables us to consider nonlinearities which grow either super quadratic or asymptotic quadratic at infinity.


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Additional Information

Yanheng Ding
Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, People’s Republic of China

Tian Xu
Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, People’s Republic of China
Address at time of publication: Center for Applied Mathematics, Tianjin University, 300072 Tianjin, People’s Republic of China
Email: xutian@amss.ac.cn

DOI: https://doi.org/10.1090/tran/6626
Keywords: Reaction-diffusion system, singular perturbation, concentration
Received by editor(s): April 29, 2014
Received by editor(s) in revised form: December 5, 2014
Published electronically: March 1, 2016
Additional Notes: The second author is the corresponding author
Dedicated: Dedicated to Antonio Ambrosetti on the occasion of his 70th birthday
Article copyright: © Copyright 2016 American Mathematical Society