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Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions. (English) Zbl 0304.35008


MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
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[1] Arima, R.; Hasegawa, S., On global solutions for mixed problems of a semilinear differential equation, (Proc. Japan Acad., 39 (1963)), 721-725 · Zbl 0173.11804
[2] Auchmuty, J. F.G, Lyapunov methods and equations of parabolic type, (Proceedings Battelle Summer Institute on Applications of Non-Linear Analysis. Proceedings Battelle Summer Institute on Applications of Non-Linear Analysis, Springer-Verlag Lecture Notes (1972)), to be published in · Zbl 0269.35057
[3] Ball, J. M., Stability theory for an extensible beam, J. Differential Equations, 14, 399-418 (1973) · Zbl 0247.73054
[4] Canosa, J., On a nonlinear diffusion equation describing population growth, IBM J. Research Development, 17, 307-313 (1973) · Zbl 0266.65080
[5] Chafee, N.; Infante, E. F., A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4, 17-37 (1974) · Zbl 0296.35046
[6] Chafee, N., A stability analysis for a semilinear parabolic partial differential equation, J. Differential Equations, 15, 522-523 (1974) · Zbl 0271.35043
[7] Dafermos, C. M., An invariance principle for compact processes, J. Differential Equations, 9, 239-252 (1971) · Zbl 0236.34038
[8] Dafermos, C. M., Applications of the invariance principle for compact processes. I. Asymptotically dynamical systems, J. Differential Equations, 9, 291-299 (1971) · Zbl 0247.34068
[9] Dafermos, C. M., Uniform processes and semicontinuous Liapunov functionals, J. Differential Equations, 11, 401-415 (1972) · Zbl 0257.35006
[10] Dafermos, C. M., Applications of the invariance principle for compact processes II. Asymptotic behavior of solutions of a hyperbolic conservation law, J. Differential Equations, 11, 416-424 (1972) · Zbl 0252.35045
[11] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall: Prentice-Hall New Jersey · Zbl 0144.34903
[12] Gelfand, I. M.; Fomin, S. V., Calculus of Variations (1963), Prentice-Hall: Prentice-Hall Englewood Cliffs, N J · Zbl 0127.05402
[13] Hale, J. K.; Infante, E. F., Extended dynamical systems and stability theory, (Proc. Nat. Acad. Sci. U.S.A., 58 (1967)), 405-409 · Zbl 0155.42301
[14] Hale, J. K., Dynamical systems and stability, J. Math. Anal. Appl., 26, 39-59 (1969) · Zbl 0179.13303
[15] Kanel, Ya. I., On the stability of solutions for a Cauchy problem involving an equation encountered in the theory of combustion, Mat. Sbornik, 59, 245-288 (1962), (In Russian) · Zbl 0152.10302
[16] Keller, H. B.; Cohen, D. S., Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16, 1361-1376 (1967) · Zbl 0152.10401
[17] Kolmogoroff, A.; Petrovsky, I.; Piscounoff, N., Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin de l’Université d’État à Moscou, Série Internationale, Vol. I (1937) · Zbl 0018.32106
[18] LaSalle, J. P., Stability theory and the asymptotic behavior of dynamical systems, (Dynamic Stability of Structures. Dynamic Stability of Structures, Proceedings of the International Conference (1966), Pergamon: Pergamon New York) · Zbl 0153.40602
[19] LaSalle, J. P., An invariance principle in the theory of stability, (Hale, J. K.; LaSalle, J. P., International Symposium on Differential Equations and Dynamical Systems (1967), Academic Press: Academic Press New York), 277-286 · Zbl 0183.09401
[20] LaSalle, J. P.; Lefschetz, S., Stability by Liapunov’s Direct Method with Applications (1961), Academic Press: Academic Press New York · Zbl 0098.06102
[21] Nagumo, J.; Arimoto, S.; Yoshizawa, S., An active pulse transmission line simulating nerve axon, (Proc. IRE, 50 (1962)), 2061-2070
[22] Nagumo, J.; Yoshizawa, S.; Arimoto, S., Bistable transmission lines, IEEE Transactions on Circuit Theory, CT-12, 400-412 (1965)
[23] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0153.13602
[24] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, 979-1000 (1972) · Zbl 0223.35038
[25] Sattinger, D. H., Topics in Stability and Bifurcation Theory (1973), Springer-Verlag: Springer-Verlag New York · Zbl 0268.35042
[26] Slemrod, M., Asymptotic behavior of a class of abstract dynamical systems, J. Differential Equations, 7, 584-600 (1970) · Zbl 0275.93023
[27] Slemrod, M.; Infante, E. F., Asymptotic stability criteria for linear systems of difference-differential equations of neutral type and their discrete analogues, J. Math. Anal. Appl., 38, 399-415 (1972) · Zbl 0202.10301
[28] Sobolev, S. L., Partial Differential Equations of Mathematical Physics (1964), Pergamon: Pergamon New York · Zbl 0123.06508
[29] Yamaguti, M., The asymptotic behavior of the solution of a semi-linear partial differential equation related to an active pulse transmission line, (Proc. Japan Acad., 39 (1963)), 726-730 · Zbl 0173.11901
[30] Yoshizawa, S., Population growth process described by a semilinear parabolic equation, Math. Biosci., 7, 291-303 (1970) · Zbl 0212.52102
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