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Keywords:
reaction-diffusion system; unilateral conditions; quasivariational inequality; Leray-Schauder degree; eigenvalue; stability
reaction-diffusion system; unilateral conditions; quasivariational inequality; Leray-Schauder degree; eigenvalue; stability
Summary:
We consider a reaction-diffusion system of the activator-inhibitor type with unilateral boundary conditions leading to a quasivariational inequality. We show that there exists a positive eigenvalue of the problem and we obtain an instability of the trivial solution also in some area of parameters where the trivial solution of the same system with Dirichlet and Neumann boundary conditions is stable. Theorems are proved using the method of a jump in the Leray-Schauder degree.
References:
[1] A. Gierer, H. Meinhardt: Biological pattern formation involving lateral inhibition. In: Some Mathematical Questions in Biology. VI. Lectures on Mathematics in the Life Sciences, vol 7, 1974, pp. 163–183. MR 0452787
[2] P. Quittner: On the principle of linearized stability for variational inequalities. Math. Ann. 283 (1989), 257–270. MR 0980597
[3] P. Drábek, M. Kučera: Reaction-diffusion systems: Destabilizing effect of unilateral conditions. Nonlinear Analysis, Theory, Methods and Applications 12 (1988), 1172–1192. MR 0969497
[4] M. Kučera, J. Neustupa: Destabilizing effect of unilateral conditions in reaction-diffusion systems. Comment. Math. Univ. Carolinae 27 (1986), 171–187. MR 0843429
[5] P. Quittner: Bifurcation points and eigenvalues of inequalities of reaction-diffusion type. J. Reine Angew. Math. 380 (1987), 1–13. MR 0916198 | Zbl 0617.35053
[6] M. Bosák, M. Kučera: Bifurcation for quasivariational inequalities of reaction-diffusion type. Stability and Appl. Anal. of Cont. Media 3 (1993), 111–127.
[7] M. Kučera: Reaction-diffusion systems: Bifurcation and stabilizing effect of conditions given by inclusions. Nonlinear Analysis, Theory, Methods and Applications 27 (1996), 249–260. DOI 10.1016/0362-546X(95)00055-Z | MR 1391435
[8] M. Kučera: Bifurcation of solutions to reaction-diffusion systems with unilateral conditions. In: Navier-Stokes Equations and Related Nonlinear Problems, A. Sequeira (ed.), Plenum Press, New York, 1995, pp. 307–322. MR 1373224
[9] J. Eisner, M. Kučera: Spatial patterns for reaction-diffusion systems with conditions described by inclusions. Appl. Math. 42 (1997), 421–449. DOI 10.1023/A:1022203129542 | MR 1475051
[10] J. Eisner, M. Kučera: Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions. (to appear). MR 1759546
[11] J. L. Lions, E. Magenes: Problèmes aux Limites non Homogènes et Applications. Dunod, Paris, 1970. MR 0291887
[12] G. Duvaut, J. L. Lions: Les Inéquations en Mécanique et en Physique. Dunod, Paris, 1972. MR 0464857
[13] E. H. Zarantonello: Projections on convex sets in Hilbert space and spectral theory. In: Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, pp. 237–424. MR 0388177 | Zbl 0281.47043
[14] M. Mimura, Y. Nishiura and M. Yamaguti: Some diffusive prey and predator systems and their bifurcation problems. Ann. N. Y. Acad. Sci. 316 (1979), 490–521. MR 0556853
[15] Y. Nishiura: Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Analysis 13 (1982), 555–593. DOI 10.1137/0513037 | MR 0661590 | Zbl 0505.76103
[16] P. Quittner: Solvability and multiplicity results for variational inequalities. Comment. Math. Univ. Carolinae 30 (1989), 281–302. MR 1014128 | Zbl 0698.49004
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