A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions
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We consider a simple reaction-diffusion system exhibiting Turing’s diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential operator, and a variational approach is used in a certain non-direct way.
Keywordsreaction-diffusion system unilateral condition variational inequality local bifurcation variational approach spatial patterns
MSC 201035B32 35K57 35J50 35J57 47J20
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- P. Drábek, M. Kučera, M. Míková: Bifurcation points of reaction-diffusion systems with unilateral conditions. Czech. Math. J. 35 (1985), 639–660.Google Scholar
- J. Eisner, M. Kučera: Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions. Fields Institute Communications 25 (2000), 239–256.Google Scholar
- M. Kučera, L. Recke, J. Eisner: Smooth bifurcation for variational inequalities and reaction-diffusion systems. Progresses in Analysis (J.G.W. Begehr, R.P. Gilbert, M.W. Wong, eds.). World Scientific, Singapore-New Jersey-London-Hong Kong, 2001, pp. 1125–1133.Google Scholar
- Y. Nishiura: Global structure of bifurcating solutions of some reaction-diffusion systems and their stability problem. Proceedings of the 5th Int. Symp. Computing Methods in Applied Sciences and Engineering, Versailles, France, 1981 (R. Glowinski, J. L. Lions, eds.). North-Holland, Amsterdam-New York-Oxford,1982.Google Scholar
- E. Zeidler: Nonlinear Functional Analysis and Its Applications, vol. III. Springer, New York-Berlin-Heidelberg, 1985.Google Scholar
- E. Zeidler: Nonlinear Functional Analysis and Its Applications, vol. IV. Springer, New York-Berlin-Heidelberg, 1988.Google Scholar