Eigenvalues of inequalities of reaction-diffusion type and destabilizing effect of unilateral conditions. (English) Zbl 0608.35032

The linearized stability of the reaction-diffusion system \[ \partial u/\partial t=d\Delta u+f(u,v);\quad \partial v/\partial t=\Delta v+g(u,v) \] with unilateral boundary conditions \(\partial v/\partial n\geq 0\), \(v\geq 0\) on part of the boundary are considered. In particular it is shown that the unilateral conditions have a destabilizing effect, compared with the corresponding bilateral conditions. This is achieved by writing the system as a variational inequality in the abstract Lions formulation.
Reviewer: S.Banks


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B35 Stability in context of PDEs
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35P05 General topics in linear spectral theory for PDEs
35K57 Reaction-diffusion equations
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[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] G. Duvant J.-L. Lions: Les inéquations en mechanique et on physique. Dunod, Paris 1972.
[3] S. Fučík A. Kufner: Nonlinear differential equations. Elsevier, Scient. Publ. Comp., Amsterdam-Oxford-New York 1980. · Zbl 0426.35001
[4] P. Drábek M. Kučera M. Míková: Bifurcation points of reaction-diffusion systems with unilateral conditions. Czechoslovak Math. J. 35 (110) 1985, 639-660. · Zbl 0604.35042
[5] P. Drábek M. Kučera: Reaction-diffusion systems: Destabilizing eifect of unilateral conditions. To appear. · Zbl 0671.35043
[6] H. Kielhöfer: Stability and semilinear evolution equations in Hilbert space. Arch. Rational Mech. Anal., 57 (1974), 150-165. · Zbl 0337.34056
[7] M. Kučera: A new method for obtaining eigenvalues of variational inequalities based on bifurcation theory. Čas. pěst. mat. 104 (1979), 389-411.
[8] 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
[9] M. Kučera: Bifurcations points of variational inequalities. Czechoslovak Math. J. 32 (107) 1982, 208-226. · Zbl 0621.49006
[10] M. Kučera: Bifurcation points of inequalities of reaction-diffusion type. To appear. · Zbl 0898.35010
[11] M. Kučera J. Neustupa: Destabilizing effect of unilateral conditions in reaction-diffusion systems. Comment. Math. Univ. Carol., 27 (1986), 171-187. · Zbl 0597.35006
[12] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Praha 1967. · Zbl 1225.35003
[13] M. Mimura Y. Nishiura M. Yamaguti: Some diffusive prey and predator systems and their bifurcation problems. Ann. New York Acad. Sci., 316 (1979), 490-521. · Zbl 0437.92027
[14] 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
[15] 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.
[16] E. Zeidler: Vorlesungen über nichtlineare Funktionalanalysis \(l\)-Fixpunktsätze. TeubnerTexte zur Mathematik, Leipzig 1976. · Zbl 0326.47053
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