×

zbMATH — the first resource for mathematics

Bifurcation points of reaction-diffusion systems with unilateral conditions. (English) Zbl 0604.35042
Stationary solutions of reaction-diffusion systems with unilateral conditions are considered. The problem is formulated in terms of abstract inequalities on cones in a Hilbert space. The diffusion coefficient plays the role of a bifurcation parameter. It is proved that there exists a bifurcation point lying in the domain of parameters for which a spatially homogeneous stationary solution of the classical problem with suitable boundary conditions is stable. It follows from here that spatially nonhomogeneous stationary solutions of the reaction-diffusion system with unilateral conditions exist for some parameters for which the problem with the classical boundary conditions has only a spatially homogeneous stationary solution.

MSC:
35K57 Reaction-diffusion equations
35B32 Bifurcations in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] E. N. Dancer: On the structure of solutions of non-linear eigenvalue problems. Ind. Univ. Math. J. 23 (1974), 1069-1076. · Zbl 0276.47051
[2] S. Fučík A. Kufner: Nonlinear differential equations. Elsevier, Scient. Publ. Comp., Amsterdam-Oxford- New York 1980. · Zbl 0426.35001
[3] P. Drábek M. Kučera: Eigenvalues of inequalities of reaction-diffusion type and destabilizing effect of unilateral conditions. To appear in Czech. Math. J., 1986. · Zbl 0608.35032
[4] M. Kučera: A new method for obtaining eigenvalues of variational inequalities based on bifurcation theory. Čas. pěst. mat. 104 (1979), 389-411.
[5] M. Kučera: A new method for obtaining eigenvalues of variational inequalities. Operators with multiple eigenvalues. Czechoslovak Math. J., 32 (107) (1982), 197-207. · Zbl 0621.49005
[6] M. Kučera: Bifurcations points of variational inequalities. Czechoslovak Math. J., 32 (107) (1982), 208-226. · Zbl 0621.49006
[7] M. Kučera J. Neustupa: Destabilizing effect of unilateral conditions in reaction-diffusion systems. To appear in Comment. Math. Univ. Carol., 1986. · Zbl 0597.35006
[8] Y. Nishiura: Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. Vol. 13, No. 4, July 1982, 555-593. · Zbl 0505.76103
[9] P. H. Rabinowitz: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971), 487-513. · Zbl 0212.16504
[10] E. H. Zarantonello: Projections on convex sets in Hilbert space and spectral theory. In ”Contributions to Nonlinear Functional Analysis” (edited by E. H. Zarantonello). Academic Press, New York, 1971. · Zbl 0281.47043
[11] E. Zeidler: Vorlesungen über nichtlineare Funktionalanalysis I - Fixpunktsätze. Teubner-Texte zur Mathematik, Leipzig 1976. · Zbl 0326.47053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.