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Invariant manifolds and bifurcation for quasilinear reaction-diffusion systems. (English) Zbl 0818.35039

A parameter-dependent strongly coupled reaction-diffusion system with nonlinear boundary conditions is considered. Existence and attractivity of some parameter-dependent locally invariant manifolds is established. The Hopf bifurcation is studied and an algorithm that allows to determine whether the supercritical or the subcritical case occurs is given.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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[1] Amann, H., Dynamic theory of quasilinear parabolic equations II. Reaction diffusion systems, Diff. Integral Eqns, 3, 13-75 (1990) · Zbl 0729.35062
[2] Amann, H., Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, (Triebel, H.; Schmeisser, H. J., Function Spaces, Differential Operators and Nonlinear Analysis (1993), Teubner: Teubner Leipzig) · Zbl 0810.35037
[4] Drangeid, A.-K., The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Analysis, 13, 1091-1113 (1989) · Zbl 0694.35009
[6] Marsden, J. E.; McCracken, M., The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, Vol. 19 (1976), Springer: Springer New York · Zbl 0346.58007
[7] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer: Springer Berlin · Zbl 0456.35001
[8] Carr, J., Applications of Centre Manifold Theory, Applied Mathematical Sciences, Vol. 35 (1981), Springer: Springer New York · Zbl 0464.58001
[9] Da Prato, G.; Grisvard, P., Equations d’évolution abstraites nonlinéaires de type parabolique, Annali Mat. pura appl., 120, 4, 329-396 (1979) · Zbl 0471.35036
[10] Prato Da, G.; Lunardi, A., Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space, Archs ration. Mech. Analysis, 101, 115-144 (1988) · Zbl 0661.35044
[11] Ruelle, D., Elements of Differentiable Dynamics and Bifurcation Theory (1989), Academic Press: Academic Press Boston · Zbl 0684.58001
[12] Amann, H., Ordinary Differential Equations. An Introduction to Nonlinear Analysis (1990), De Gruyter: De Gruyter Berlin
[13] Chow, S. N.; Hale, J. K., Methods of Bifurcation Theory (1982), Springer: Springer Berlin
[14] Iooss, G., Bifurcation of Maps and Applications (1979), North Holland: North Holland Amsterdam · Zbl 0408.58019
[15] Crandall, M. G.; Rabinowitz, P., The Hopf bifurcation theorem in infinite dimensions, Archs ration. Mech. Analysis, 67, 53-72 (1977/78) · Zbl 0385.34020
[16] Ize, J., Periodic solutions of nonlinear parabolic equations, Communs partial diff. Eqns, 4, 1299-1387 (1979) · Zbl 0436.35012
[17] Kielhöfer, H., Hopf bifurcation at multiple eigenvalues, Archs ration. Mech. Analysis, 69, 53-83 (1979) · Zbl 0398.34058
[18] Kielhöfer, H., Generalized Hopf bifurcation in Hilbert space, Math. Meth. Appl. Sci., 1, 498-513 (1979) · Zbl 0451.34059
[19] Amann, H., Hopf bifurcation in quasilinear reaction diffusion systems, (Delay Differential Equations and Dynamical Systems, Proceedings. Delay Differential Equations and Dynamical Systems, Proceedings, Claremont 1990 (1991), Springer: Springer Berlin) · Zbl 0780.35051
[20] Da Prato, G.; Lunardi, A., Hopf bifurcation for fully nonlinear equations in Banach spaces, Ann. Inst. H. Poincaré, 3, 315-329 (1986) · Zbl 0609.34066
[21] Amann, H., Dynamic theory of quasilinear parabolic equations—I. Abstract evolution equations, Nonlinear Analysis, 12, 895-919 (1988) · Zbl 0666.35043
[22] Angenent, S. B., Nonlinear analytic semiflows, Proc. R. Soc. Edinb., 115A, 91-107 (1990) · Zbl 0723.34047
[23] Lunardi, A., Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations, Math. Nachr., 121, 295-318 (1985) · Zbl 0568.47035
[24] Simonett, G., Zentrumsmannigfaltigkeiten für quasilinear parabolische Gleichungen, Institut für Angewandte Analysis und Stochastik. Institut für Angewandte Analysis und Stochastik, Report No. 2 (1992), Berlin
[25] Simonett, G., Quasilinear parabolic equations and semiflows, (Proc. Semigroup Theory and Evolution Equations, Lecture Notes in Pure Applied Mathematics, Vol. 155 (1994), Dekker: Dekker New York), 523-536 · Zbl 0830.35010
[26] Daners, D.; Koch Medina, P., Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series, Vol. 279 (1994), Longman: Longman New York
[27] Kato, T., Perturbation Theory for Linear Operators (1966), Springer: Springer New York · Zbl 0148.12601
[29] Amann, H., Parabolic evolution equations in interpolation and extrapolation spaces, J. funct. Analysis, 78, 233-270 (1988) · Zbl 0654.47019
[30] Bergh, J.; Löfström, J., Interpolation Spaces. An Introduction (1976), Springer: Springer Berlin · Zbl 0344.46071
[31] Triebel, H., Interpolation Theory, Function Spaces and Differential Operators (1978), North Holland: North Holland Amsterdam · Zbl 0387.46032
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