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Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations


Author: Klaus W. Schaaf
Journal: Trans. Amer. Math. Soc. 302 (1987), 587-615
MSC: Primary 35R10; Secondary 35B40, 35K55
MathSciNet review: 891637
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Abstract: This paper is a generalization of the theory of the KPP and bistable nonlinear diffusion equations. It is shown that traveling wave solutions exist for nonlinear parabolic functional differential equations (FDEs) which behave very much like the well-known solutions of the classical KPP and bistable equations. Among the techniques used are maximum principles, sub- and supersolutions, phase plane techniques for FDEs and perturbation of linear operators.


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  • [1] D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, Nonlinear diffusion (NSF-CBMS Regional Conf. Nonlinear Diffusion Equations, Univ. Houston, Houston, Tex., 1976) Pitman, London, 1977, pp. 1–23. Res. Notes Math., No. 14. MR 0490046
  • [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Springer, Berlin, 1975, pp. 5–49. Lecture Notes in Math., Vol. 446. MR 0427837
  • [3] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, 10.1016/0001-8708(78)90130-5
  • [4] C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 2, 315–330. MR 0416645
  • [5] Maury Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc. 44 (1983), no. 285, iv+190. MR 705746, 10.1090/memo/0285
  • [6] Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338
  • [7] Hirsh Cohen, Nonlinear diffusion problems, Studies in applied mathematics, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N. J.), 1971, pp. 27–64. MAA Studies in Math., Vol. 7. MR 0447846
  • [8] O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol. 6 (1978), no. 2, 109–130. MR 647282, 10.1007/BF02450783
  • [9] -, Run for your life. A note on the asymptotic speed of propagation of an epidemic, Math. Centre, Amsterdam, Report TW 176/78, 1978.
  • [10] Paul C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, vol. 28, Springer-Verlag, Berlin-New York, 1979. MR 527914
  • [11] Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Bull. Amer. Math. Soc. 81 (1975), no. 6, 1076–1078. MR 0380111, 10.1090/S0002-9904-1975-13922-X
  • [12] Paul C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rational Mech. Anal. 75 (1980/81), no. 4, 281–314. MR 607901, 10.1007/BF00256381
  • [13] Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
  • [14] I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142
  • [15] K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol. 2 (1975), no. 3, 251–263. MR 0411693
  • [16] Jack Hale, Theory of functional differential equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. MR 0508721
  • [17] Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • [18] Willi Jäger, Hermann Rost, and Petre Tautu (eds.), Biological growth and spread, Lecture Notes in Biomathematics, vol. 38, Springer-Verlag, Berlin-New York, 1980. Mathematical theories and applications. MR 609340
  • [19] Yoshinori Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka J. Math. 13 (1976), no. 1, 11–66. MR 0422875
  • [20] Ja. I. Kanel′, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb. (N.S.) 59 (101) (1962), no. suppl., 245–288 (Russian). MR 0157130
  • [21] -, On the stability of solutions of the equation of combustion theory for finite initial functions, Mat. Sb. (N.S.) 65(107) (1964), 398-413.
  • [22] T. Kato, Perturbation theory for linear operators, Spinger-Verlag, Berlin and New York, 1980.
  • [23] Kusuo Kobayashi, On the semilinear heat equations with time-lag, Hiroshima Math. J. 7 (1977), no. 2, 459–472. MR 0454429
  • [24] A. N. Kolmogoroff, I. G. Petrovski, and N. S. Piskunoff, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Bull. Math., Série Internat., Sec. A, Math. et Méc. 1(6) (1937), 1-25.
  • [25] H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 28 (1975), no. 3, 323–331. MR 0400428
  • [26] Ray Redheffer and Wolfgang Walter, Das Maximumprinzip in unbeschränkten Gebieten für parabolische Ungleichungen mit Funktionalen, Math. Ann. 226 (1977), no. 2, 155–170 (German). MR 0450786
  • [27] Franz Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), no. 3-4, 213–234. MR 516224, 10.1017/S0308210500010258
  • [28] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 0435602
  • [29] D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Differential Equations 25 (1977), no. 1, 130–144. MR 0447813
  • [30] Konrad Schumacher, Travelling-front solutions for integro-differential equations. I, J. Reine Angew. Math. 316 (1980), 54–70. MR 581323, 10.1515/crll.1980.316.54
  • [31] Konrad Schumacher, Travelling-front solutions for integro-differential equations. II, Biological growth and spread (Proc. Conf., Heidelberg, 1979) Lecture Notes in Biomath., vol. 38, Springer, Berlin-New York, 1980, pp. 296–309. MR 609368
  • [32] J. Szarski, Strong maximum principle for non-linear parabolic differential-functional inequalities in arbitrary domains, Ann. Polon. Math. 31 (1975), no. 2, 197–203. MR 0412643
  • [33] Horst R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math. 306 (1979), 94–121. MR 524650, 10.1515/crll.1979.306.94
  • [34] Kōhei Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (1978), no. 3, 453–508. MR 509494
  • [35] Wolfgang Walter, Differential and integral inequalities, Translated from the German by Lisa Rosenblatt and Lawrence Shampine. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York-Berlin, 1970. MR 0271508
  • [36] H. F. Weinberger, Asymptotic behavior of a class of discrete-time models in population genetics, Applied nonlinear analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978) Academic Press, New York-London, 1979, pp. 407–422. MR 537552
  • [37] E. M. Wright, The linear difference-differential equation with asymptotically constant coefficients, Amer. J. Math. 70 (1948), 221–238. MR 0025063

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DOI: https://doi.org/10.1090/S0002-9947-1987-0891637-2
Article copyright: © Copyright 1987 American Mathematical Society