The approach of solutions of nonlinear diffusion equations to travelling front solutions. (English) Zbl 0361.35035


35K55 Nonlinear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI


[1] D.G. Aronson & H.F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation, in Partial Differential Equations and Related Topics, ed. J.A. Goldstein. Lecture Notes in Mathematics 446, 5–49 New York: Springer 1975. · Zbl 0325.35050
[2] D.G. Aronson & H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., to appear. · Zbl 0407.92014
[3] G.I. Barenblatt and Ya.B. Zel’dovič, Intermediate asymptotics in mathematical physics, Usp. Mat. Nauk 26, 115–129 (1971); Russian Math. Surveys 26, 45–61 (1971).
[4] K.-N. Chueh, On the asymptotic behavior of solutions of semilinear parabolic partial equations, Ph.D. Thesis, University of Wisconsin, 1975.
[5] H. Cohen, Nonlinear diffusion problems, in Studies in Applied Mathematics, ed. A.H. Taub, Studies in Mathematics No. 7, Math. Assoc. of America and Prentice-Hall, 27–64 (1971).
[6] P.C. Fife, Pattern formation in reacting and diffusing systems, J. Chem. Phys. 64, 554–564 (1976). · doi:10.1063/1.432246
[7] P.C. Fife, Singular perturbation and wave front techniques in reaction-diffusion problems, Proc. AMS-SIAM Symposium on Asymptotic Methods and Singular Perturbations, New York, 1976 · Zbl 0372.35006
[8] P.C. Fife & J.B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Bull. Amer. Math. Soc. 81, 1075–1078 (1975). · Zbl 0318.35046 · doi:10.1090/S0002-9904-1975-13922-X
[9] R.A. Fisher, The advance of advantageous genes, Ann. of Eugenics 7, 355–369 (1937). · JFM 63.1111.04 · doi:10.1111/j.1469-1809.1937.tb02153.x
[10] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall: Englewood Cliffs, N.J. 1964. · Zbl 0144.34903
[11] I.M. Gelfand, Some problems in the theory of quasilinear equations, Usp. Mat. Nauk. (N.S.) 14, 87–158 (1959); A.M.S. Translations (2) 29, 295–381 (1963).
[12] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics, and Epidemics, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1975. · Zbl 0304.92012
[13] Y. Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka J. Math. 13, 11–66 (1976). · Zbl 0344.35050
[14] Ya.I. Kanel’, On the stabilization of solutions of the Cauchy problem for equations arising in the theory of combustion, Mat. Sbornik 59, 245–288 (1962). See also Dokl. Akad. Nauk SSSR 132, 268–271 (1960), (= Soviet Math. Dokl. 1, 533–536 (1960)) and Dokl Akad. Nauk SSSR 136, 277–280 (1961) (= Soviet Math. Dokl. 2, 48–51 (1961)).
[15] Ya.I. Kanel’, On the stabilization of solutions of the equations of the theory of combustion with initial data of compact support, Mat. Sbornik 65, 398–413 (1964).
[16] A.N. Kolmogorov, I.G. Petrovskii, & N.S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ. 1:7, 1–26 (1937).
[17] H.P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 28, 323–331 (1975). · Zbl 0316.35053 · doi:10.1002/cpa.3160280302
[18] E.W. Montroll, Nonlinear rate processes, especially those involving competitive processes, in Statistical Mechanics, ed. Rice, Freed, and Light, Univ. of Chicago Press, 69–89 (1972).
[19] J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Eng. 50, 2061–2070 (1962).
[20] M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall: Englewood Cliffs, N.J. 1967. · Zbl 0153.13602
[21] F. Rothe, Über das asymptotische Verhalten der Lösungen einer nichtlinearen parabolischen Differentialgleichung aus der Populationsgenetik, Ph.D. Dissertation, University of Tübingen, 1975.
[22] D.H. Sattinger, Weighted norms for the stability of travelling waves, J. Differential Equations, to appear. · Zbl 0315.35010
[23] A.N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosciences 31, 307–315 (1976). · Zbl 0333.35048 · doi:10.1016/0025-5564(76)90087-0
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.