Abstract
The multiple existences and their stability properties of stationary solutions with a single transition layer in some scalar reaction-diffusion equation are shown. Each solution is constructed by using classical singular perturbation methods and its stability property is determined by a simple algebraic quantity, say index, appearing in the construction of a solution.
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Angenent, S. B., Mallet-Paret, J., and Peletier, L. A. (1987). Stable transition layers in semilinear boundary value problem.J. Diff. Eq. 67, 212–242.
Coddington, E. A., and Levinson, N. (1955).Theory of Ordinary Differential Equations, McGraw-Hill, New York.
Eckhaus, W. (1979).Asymptotic Analysis of Singular Perturbation, North-Holland, Amsterdam.
Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations.J. Diff. Eq. 31, 53–98.
Fife, P. C. (1976). Boundary and interior transition layer phenomena for pairs of second-order differential equations,J. Math. Anal. Appl. 54, 497–521.
Gardner, R., and Jones, C. K. R. T. (1991). A stability index for steady state solutions of boundary value problems for parabolic systems.J. Diff. Eq. 91, 181–203.
Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems, Math. Surv. Monogr. 25, Am. Math. Soc.
Hale, J. K., and Sakamoto, K. (1988). Existence and stability of transition layers,Jap. J. Appl. Math. 5, 367–405.
Ikeda, H., and Mimura, M. (1991). Stability analysis of stationary solutions of bistable reaction-variable diffusion systems.SIAM J. Math. Anal. 22, 1651–1678.
Ito, M. (1985). A remark on singular perturbation methods.Hiroshima Math. J. 14, 619–629.
Kurland, H. L. (1983). Monotone and oscillatory equilibrium solutions of a problem arising in population genetics. InContemporary Mathematics, Vol. 17, Am. Math. Soc., Providence, RI, pp. 323–342.
Maginu, K. (1984). A geometrical condition for the instability of solitary travelling wave solutions in reaction-diffusion equations. Preprint.
Mimura, M., Tabata, M., and Hosono, Y. (1981). Multiple solutions of two point boundary value problems of Neumann type with a small parameter.SIAM J. Math. Anal. 11, 613–631.
Nishiura, Y., and Fujii, H. (1987). Stability of singularly perturbed solutions to systems of reaction-diffusion equations.SIAM J. Math. Anal. 18, 1726–1770.
Nishiura, M., and Tsujikawa, T. (1990). Instability of singularly perturbed Neumann layer solutions in reaction-diffusion systems.Hiroshima Math. J. 20, 297–329.
Rocha, C. (1988). Examples of attractors in scalar reaction-diffusion equations,J. Diff. Eq. 73, 178–195.
Rocha, C. (1991). Properties of the attractor of a scalar parabolic PDE.J. Dyn. Diff. Eq. 3, 575–591.
Sakamoto, K. (1990). Construction and stability analysis of transition layer solutions in reaction-diffusion systems.Tohoku Math. J. 42, 17–44.
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Ikeda, H. Stability characteristics of transition layer solutions. J Dyn Diff Equat 5, 625–671 (1993). https://doi.org/10.1007/BF01049142
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DOI: https://doi.org/10.1007/BF01049142