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

Schaaf, Klaus W.

doi:10.1090/S0002-9947-1987-0891637-2

Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations
HTML articles powered by AMS MathViewer

by Klaus W. Schaaf PDF

Trans. Amer. Math. Soc. 302 (1987), 587-615 Request permission

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.

References

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) Res. Notes Math., No. 14, Pitman, London, 1977, pp. 1–23. MR 0490046
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) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 5–49. MR 0427837
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, DOI 10.1016/0001-8708(78)90130-5
C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 2, 315–330. MR 416645, DOI 10.1017/S0305004100052944
Maury Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc. 44 (1983), no. 285, iv+190. MR 705746, DOI 10.1090/memo/0285
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
Hirsh Cohen, Nonlinear diffusion problems, Studies in applied mathematics, MAA Studies in Math., Vol. 7, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N. J.), 1971, pp. 27–64. MR 0447846
O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol. 6 (1978), no. 2, 109–130. MR 647282, DOI 10.1007/BF02450783

Run for your life. A note on the asymptotic speed of propagation of an epidemic

Paul C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, vol. 28, Springer-Verlag, Berlin-New York, 1979. MR 527914, DOI 10.1007/978-3-642-93111-6
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 380111, DOI 10.1090/S0002-9904-1975-13922-X
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, DOI 10.1007/BF00256381
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol. 2 (1975), no. 3, 251–263. MR 411693, DOI 10.1007/BF00277154
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721, DOI 10.1007/978-1-4612-9892-2
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
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
Yoshinori Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka Math. J. 13 (1976), no. 1, 11–66. MR 422875
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

On the stability of solutions of the equation of combustion theory for finite initial functions

65(107)

Perturbation theory for linear operators

Kusuo Kobayashi, On the semilinear heat equations with time-lag, Hiroshima Math. J. 7 (1977), no. 2, 459–472. MR 454429

Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique

1(6)

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 400428, DOI 10.1002/cpa.3160280302
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 450786, DOI 10.1007/BF01360866
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, DOI 10.1017/S0308210500010258
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 435602, DOI 10.1016/0001-8708(76)90098-0
D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Differential Equations 25 (1977), no. 1, 130–144. MR 447813, DOI 10.1016/0022-0396(77)90185-1
Konrad Schumacher, Travelling-front solutions for integro-differential equations. I, J. Reine Angew. Math. 316 (1980), 54–70. MR 581323, DOI 10.1515/crll.1980.316.54
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
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 412643, DOI 10.4064/ap-31-2-197-203
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, DOI 10.1515/crll.1979.306.94
K\B{o}hei 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, DOI 10.1215/kjm/1250522506
Wolfgang Walter, Differential and integral inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York-Berlin, 1970. Translated from the German by Lisa Rosenblatt and Lawrence Shampine. MR 0271508, DOI 10.1007/978-3-642-86405-6
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
E. M. Wright, The linear difference-differential equation with asymptotically constant coefficients, Amer. J. Math. 70 (1948), 221–238. MR 25063, DOI 10.2307/2372324