Stability and Spectral Comparison of a Reaction–Diffusion System with Mass Conservation
- 160 Downloads
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.
KeywordsReaction diffusion system Mass conservation Cell polarity Global-in-time behavior Lyapunov function Spectral comparison
Mathematics Subject Classification35K457 92C37.1
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.
- 5.Fix, G.J.: Phase field methods for free boundary problems. In: Fasano, A., Primicerio, M. (eds.) Free Boundary Problems: Theory and Applications, pp. 580–589. Pitman, London (1983)Google Scholar
- 6.Hale, J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence (1989)Google Scholar
- 7.Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math, vol. 840. Springer, Berlin (1981)Google Scholar
- 18.Pawłov, I., Suzuki, T., Tasaki, S.: Stationary solutions to a strain-gradient type thermovisocelastic system. Differ. Integral Equ. 25, 289–340 (2012)Google Scholar
- 20.Rothe, F.: Global Solutions of Reaction–Diffusion Equations. Lecture Notes in Math, vol. 1072. Springer, Berlin (1984)Google Scholar