Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; Turing instability
reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; Turing instability
Summary:
Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the classical case without unilateral obstacles. The study is based on a variational approach to a non-variational problem which even after transformation to a variational one has an unusual structure for which usual variational methods do not apply.
References:
[1] Baltaev, J. I., Kučera, M., Väth, M.: A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions. Appl. Math., Praha 57 (2012), 143-165. DOI 10.1007/s10492-012-0010-2 | MR 2899729 | Zbl 1249.35020
[2] Drábek, P., Kučera, M., Míková, M.: Bifurcation points of reaction-diffusion systems with unilateral conditions. Czech. Math. J. 35 (1985), 639-660. MR 0809047 | Zbl 0604.35042
[3] Drábek, P., Kufner, A., Nicolosi, F.: Quasilinear Elliptic Equations with Degenerations and Singularities. De Gruyter Series in Nonlinear Analysis and Applications 5 Walter de Gruyter, Berlin (1997). MR 1460729 | Zbl 0894.35002
[4] Edelstein-Keshet, L.: Mathematical Models in Biology. The Random House/Birkhäuser Mathematics Series Random House, New York (1988). MR 1010228 | Zbl 0674.92001
[5] Eisner, J.: Critical and bifurcation points of reaction-diffusion systems with conditions given by inclusions. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 46 (2001), 69-90. DOI 10.1016/S0362-546X(99)00446-0 | MR 1845578 | Zbl 0980.35029
[6] Eisner, J., Kučera, M.: Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions. Operator Theory and Its Applications Proc. of the Int. Conf., Winnipeg, Canada, 1998 Fields Inst. Commun. 25, American Mathematical Society, Providence 239-256 (2000). MR 1759546 | Zbl 0969.35019
[7] Eisner, J., Kučera, M., Recke, L.: Direction and stability of bifurcation branches for variational inequalities. J. Math. Anal. Appl. 301 (2005), 276-294. DOI 10.1016/j.jmaa.2004.07.021 | MR 2105671 | Zbl 1058.49005
[8] Eisner, J., Kučera, M., Recke, L.: Smooth bifurcation branches of solutions for a Signorini problem. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 1853-1877. DOI 10.1016/j.na.2010.10.058 | MR 2764386 | Zbl 1213.35233
[9] Eisner, J., Kučera, M., Väth, M.: Global bifurcation of a reaction-diffusion system with inclusions. Z. Anal. Anwend. 28 (2009), 373-409. DOI 10.4171/ZAA/1390 | MR 2550696
[10] Eisner, J., Kučera, M., Väth, M.: Bifurcation points for a reaction-diffusion system with two inequalities. J. Math. Anal. Appl. 365 (2010), 176-194. DOI 10.1016/j.jmaa.2009.10.037 | MR 2585089 | Zbl 1185.35074
[11] Eisner, J., Väth, M.: Location of bifurcation points for a reaction-diffusion system with Neumann-Signorini conditions. Adv. Nonlinear Stud. 11 (2011), 809-836. DOI 10.1515/ans-2011-0403 | MR 2868433 | Zbl 1258.35020
[12] Fučík, S., Kufner, A.: Nonlinear Differential Equations. Studies in Applied Mechanics 2 Elsevier Scientific Publishing Company, Amsterdam (1980). MR 0558764
[13] Jones, D. S., Sleeman, B. D.: Differential Equations and Mathematical Biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series Chapman & Hall/CRC, Boca Raton (2003). MR 1967145 | Zbl 1020.92001
[14] Kučera, M.: Stability and bifurcation problems for reaction-diffusion systems with unilateral conditions. Differential Equations and Their Applications Equadiff 6, Proc. 6th Int. Conf. Brno, 1985 Lect. Notes Math. 1192 (1986), 227-234. DOI 10.1007/BFb0076074 | MR 0877129 | Zbl 0643.35050
[15] Kučera, M.: Reaction-diffusion systems: stabilizing effect of conditions described by quasivariational inequalities. Czech. Math. J. 47 (1997), 469-486. DOI 10.1023/A:1022411501260 | MR 1461426 | Zbl 0898.35010
[16] Kučera, M., Recke, L., Eisner, J.: Smooth bifurcation for variational inequalities and reaction-diffusion systems. Progress in Analysis. Vol. I, II Proc. of the 3rd Int. Congress of the Int. Society for Analysis, Its Applications and Computation, Berlin, 2001 World Sci. Publ., River Edge (2003), 1125-1133. MR 2032793
[17] Kučera, M., Väth, M.: Bifurcation for a reaction-diffusion system with unilateral and {Neumann} boundary conditions. J. Differ. Equations 252 (2012), 2951-2982. DOI 10.1016/j.jde.2011.10.016 | MR 2871789 | Zbl 1237.35013
[18] Mimura, M., Nishiura, Y., Yamaguti, M.: Some diffusive prey and predator systems and their bifurcation problems. Bifurcation Theory and Applications in Scientific Disciplines, Pap. Conf., New York, 1977 Ann. New York Acad. Sci. 316 (1979), 490-510. DOI 10.1111/j.1749-6632.1979.tb29492.x | MR 0556853 | Zbl 0437.92027
[19] Murray, J. D.: Mathematical Biology. Biomathematics 19 Springer, Berlin (1993). MR 1239892 | Zbl 0779.92001
[20] Nishiura, Y.: Global structure of bifurcating solutions of some reaction-diffusion systems and their stability problem. Computing Methods in Applied Sciences and Engineering V Proc. 5th Int. Symp., Versailles, 1981 North Holland, Amsterdam (1982), 185-204. MR 0784643 | Zbl 0505.76103
[21] Quittner, P.: Bifurcation points and eigenvalues of inequalities of reaction-diffusion type. J. Reine Angew. Math. 380 (1987), 1-13. MR 0916198 | Zbl 0617.35053
[22] Recke, L., Eisner, J., Kučera, M.: Smooth bifurcation for variational inequalities based on the implicit function theorem. J. Math. Anal. Appl. 275 (2002), 615-641. DOI 10.1016/S0022-247X(02)00272-X | MR 1943769 | Zbl 1018.34042
[23] Väth, M.: A disc-cutting theorem and two-dimensional bifurcation of a reaction-diffusion system with inclusions. Cubo 10 (2008), 85-100. MR 2467201 | Zbl 1169.35352
[24] Väth, M.: New beams of global bifurcation points for a reaction-diffusion system with inequalities or inclusions. J. Differ. Equations 247 (2009), 3040-3069. DOI 10.1016/j.jde.2009.09.004 | MR 2569857 | Zbl 1188.35019
[25] Väth, M.: Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities. Math. Bohem. 139 (2014), 195-211. MR 3238834 | Zbl 1340.35145
[26] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization. Springer, New York (1985). MR 0768749 | Zbl 0583.47051
[27] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. IV: Applications to Mathematical Physics. Springer, New York (1988). MR 0932255 | Zbl 0648.47036
Search
Partner of
