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Generic properties of stationary state solutions of reaction-diffusion equations. (English) Zbl 0544.34019

The authors consider stationary solutions of the equation \(u''+f(u)=0,\) with homogeneous Dirichlet or Neumann boundary conditions. They prove that the ”time map” \(\eta \to T(\eta)\) is generically a Morse function. A simpler proof was also given by the reviewer and A. Wasserman [ibid. 52, 432-438 (1984; Zbl 0488.58015)].
Reviewer: J.Smoller

MSC:

34B99 Boundary value problems for ordinary differential equations
37D15 Morse-Smale systems

Citations:

Zbl 0488.58015
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References:

[1] Abraham, R; Robbin, J, Transversal mappings and flows, (1967), Benjamin New York · Zbl 0171.44404
[2] Carr, J, Applications of centre manifold theory, () · Zbl 0464.58001
[3] Chow, S.N; Mallet-Paret, J, Integral averaging and bifurcation, J. differential equations, 26, 112-159, (1977) · Zbl 0367.34033
[4] Conley, C, Isolated invariant sets and the Morse index, N.S.F., C.B.M.S., ()
[5] {\scC. Conley and J. Smoller}, Remarks on the stability of steady-state solutions of reaction-diffusion equations, preprint. · Zbl 0458.76078
[6] Foias, C; Temam, R, Structure of the set of stationary solutions of the Navier Stokes equations, Comm. pure appl. math., 30, 149-164, (1977) · Zbl 0335.35077
[7] Foias, C; Temam, R, Remarques sur LES équations de Navier-Stokes stationnaires et LES phénomènes successifs de bifurcation, Ann. scuola norm. sup. Pisa, 5, 29-63, (1978) · Zbl 0384.35047
[8] Hale, J.K, Theory of functional differential equations, (1977), Springer-Verlag New York/Berlin · Zbl 0425.34048
[9] Hale, J.K, Ordinary differential equations, (1969), McGraw-Hill New York · Zbl 0186.40901
[10] Hale, J.K; Massatt, R, Asymptotic behavior of gradient-like systems, () · Zbl 0542.34027
[11] Henry, D, Geometry theory of semilinear parabolic equation, ()
[12] Mallet-Paret, J, Generic periodic solutions of functional differential equations, J. differential equations, 25, 163-183, (1977) · Zbl 0358.34078
[13] Matano, H, Convergence of solutions of one dimensional semilinear parabolic equations, J. math. Kyoto univ., 18, 221-227, (1978) · Zbl 0387.35008
[14] Smoller, J; Tromba, A; Wasserman, A, Nondegenerate solutions of boundary value problems, J. nonlinear anal., 4, 207-215, (1980) · Zbl 0429.34024
[15] {\scJ. Smoller and A. Wasserman}, Global bifurcation of steady-state solutions, preprint. · Zbl 0425.34028
[16] Uhlenbeck, K, Eigenfunctions of Laplace operators, Bull. amer. math. soc., 78, 1073-1076, (1972) · Zbl 0275.58003
[17] Uhlenbeck, K, Generic properties of eigenfunctions, Amer. J. math., 98, 1059-1078, (1976) · Zbl 0355.58017
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