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Convergence to travelling fronts in semilinear parabolic equations

Published online by Cambridge University Press:  14 November 2011

Franz Rothe
Franz Rothe
Affiliation:
Lehrstuhl für Biomathematik, Universität Tubingen
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Synopsis

We study the convergence to the stationary state for the parabolic equation u, = uxx + F(u). There exist wave-type solutions u(x, t) = φ(x − ct) for a continuum of velocities c. In the asymptotic behavior of this equation was investigated for a step function as initial data. In this paper we obtain the asymptotic behavior for a large class of monotone initial data.

All solutions with initial data in this class evolve to wave-type solutions, where the rate of decay of the initial data determines the asymptotic speed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 3-4 , 1978 , pp. 213 - 234
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Aronson, D. G. and Weinberger, H. F. Nonlinear diffusion in population genetics, combustion and nerve propagation. Proc. Tulane Program partial differential equations. Lecture Notes in Mathematics 446 (Berlin: Springer, 1975).Google Scholar
2Aronson, D. G. and Weinberger, H. F. Multidimensional nonlinear-diffusion arising in population genetics, to appear.Google Scholar
3Chafee, N.A stability analysis for a semilinear parabolic partial differential equation. J. Differential Equations 15 (1974), 522–540.CrossRefGoogle Scholar
4Crow, J. F. and Kimura, M.An introduction to population genetics (New York: Harper, 1970).Google Scholar
5Fife, P. C. and McLeod, J. B.The approach of solutions of nonlinear diffusion equations to travelling wave solutions. Bull. Amer. Math. Soc. 8 (1975), 1076–1078.Google Scholar
6Fisher, R. A.The genetical theory of natural selection (Oxford: Univ. Press, 1930).Google Scholar
7Fisher, R. A.The advance of advantageous genes. Ann. Eugenics 7 (1937), 355–369.Google Scholar
8Gelfand, I. M.Some problems in the theory of quasilinear equations. Uspehi Mat. Nauk. 14 (1959), 87–158 and Amer. Math. Soc. Transl. 29 (1963), 295–381.Google Scholar
9Hadeler, K. P.Mathematik für Biologen (Berlin: Springer, 1974).Google Scholar
10Hadeler, K. P. Travelling population fronts. In Population Genetics and Ecology, 585–592. (New York: Academic Press, 1976).Google Scholar
11Hadeier, K. P. and Rothe, F.Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2 (1975), 251–263.Google Scholar
12Hadeler, K. P. and Poulsen, E. T. The asymptotic displanement of a travelling front. Aarhus Univ. preprint.Google Scholar
13Hoppensteadt, F. C.Mathematical theories of populations, demographics, genetics and epidemics (Pensacola: Univ. West Florida, 1974).Google Scholar
14Kametaka, Y.On the nonlinear diffusion equation of Kolmogoroff-Petrovskii-Piskunov type. Osaka J. Math. 13 (1976), 11–66.Google Scholar
15Kanel', I. Ja.The behaviour of solutions of the Cauchy problem when time tends to infinity, in the case of quasilinear equations arising in the theory of combustion. Doki. Akad. Nauk SSSR 132 (1961), 268–271. Transl. Soviet Math. Dokl. 1, 533–536.Google Scholar
16Kanel', I. Ja.Certain problems on equations in the theory of burning. Doki. Akad. Nauk SSSR 136 (1961), 277–280 transl. Soviet Math. Dokl. 2 48–51.Google Scholar
17Kanel', I. Ja.Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory. Mat. Sb. 59 (1962), 245–288.Google Scholar
18Kanel', I. Ja.On the stability of solutions of the equation of combustion theory for finite initial functions. Mat. Sb. 107 (1964), 398–413.Google Scholar
19Kendall, D. G. Mathematical Models of the spread of infection. In, Mathematics and computer science in biology and medicine (London: Medical Res. Council, 1965).Google Scholar
20Kolmogoroff, A., Petrovsky, I. and Piscounoff, N.Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. État Moscou Ser. Internat. A Math. Méc. 1(6) (1937), 1–25.Google Scholar
21McKean, H. P.Application of Brownian Motion to the equation of Kolmogoroff-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28 (1975), 323–331.Google Scholar
22Rothe, F.Über das asymptotische Verhalten der Lösungen einer nichtlinearen parabolischen Differentialgleichung aus der Populationsgenetik (Tubingen Univ. Dissertation, 1975).Google Scholar
23Stokes, A. M.On two types of moving front in quasi-linear diffusion. Math. Biosci. 31 (1976), 307–315.Google Scholar
24Walter, W. Differential and integral inequalities. Ergebnisse der Mathematik und ihrer Grenzgebiete 55 (Berlin: Springer, 1970).Google Scholar