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Stability and Spectral Comparison of a Reaction–Diffusion System with Mass Conservation

  • Evangelos Latos
  • Yoshihisa Morita
  • Takashi Suzuki
Article

Abstract

We study the global-in-time behavior of solutions to a reaction–diffusion system with mass conservation, as proposed in the study of cell polarity, particularly, the second model of the work by Otsuji et al. (PLoS Comput Biol 3:e108, 2007). First, we show the existence of a Lyapunov function and confirm the global-in-time existence of the solution with compact orbit. Then we study the stability and instability of stationary solutions by using the semi-unfolding-minimality property and the spectral comparison. As a result the dynamics near the stationary solutions is qualitatively characterized by a variational function.

Keywords

Reaction diffusion system Mass conservation Cell polarity Global-in-time behavior Lyapunov function Spectral comparison 

Mathematics Subject Classification

35K457 92C37.1 

Notes

Acknowledgements

The authors would like to thank the referee for careful reading of the manuscript and the valuable comments for the improvement of the first version. The first author was supported from DFG Project CH 955/3-1. The second author was partially supported by the Grand-in-Aid for Scientific Research (A) No. 26247013, (B) No. 26287025 and Challenging Exploratory Research No. 24654044, Japan Society for the Promotion of Science. The third author was partially supported by JST-CREST and JSPS Core to Core Project.

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Copyright information

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Authors and Affiliations

  1. 1.University of MannheimMannheimGermany
  2. 2.Department of Applied Mathematics and InformaticsRyukoku UniversitySeta OtsuJapan
  3. 3.Center for Mathematical Modeling and Data ScienceOsaka UniversityMachikaneyamachoJapan

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