Hostname: page-component-6bf8c574d5-b4m5d Total loading time: 0 Render date: 2025-02-25T22:21:54.191Z Has data issue: false hasContentIssue false
  • English
  • Français

The travelling wave fronts of nonlinear reaction–diffusion systems via Friedlin's stochastic approaches

Published online by Cambridge University Press:  14 November 2011

Huaizhong Zhao
Huaizhong Zhao
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, England, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the asymptotic behaviour of reaction–diffusion systems with a small parameter by using the n-dimensional Feynman–Kac formula and large deviation theory. The generalised solutions are introduced in Section 2. We obtain the travelling wave joining an unstable steady state and an asymptotically stable steady state of a diffusionless dynamical system in a reaction–diffusion system with nonlinear ergodic interactions, and a special case with nonlinear reducible interactions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 2 , 1994 , pp. 273 - 299
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aronson, D. G.. The asymptotic speed of propagation of a simple epidemic. In Nonlinear Diffusion, eds Fitzgibbon, W. E. (III) and Walker, H. F., Research Notes in Mathematics 14, 1–23 (London: Pitman, 1977).Google Scholar
2Babbit, D. G.. Wiener integral representations for certain semigroups which have infinitesimal generators with matrix coefficients. J. Math. Mech. 19 (1970), 1051–1067.Google Scholar
3Britton, N. F.. Reaction–Diffusion Equations and Their Applications to Biology (New York: Academic Press, 1986).Google Scholar
4Bronsard, L. and Kohn, R. V.. Motion by mean curvature as the singular point of Ginzburgh–Landau dynamics. J. Differential Equations 90 (1991), 211–237.CrossRefGoogle Scholar
5Da, G. Prato and Zabczyk, J.. Smoothing properties of transition subgroups in Hilbert spaces. Stochastics Stochastics Rep. 35 (1991), 63–77.Google Scholar
6Deuschel, J.-D. and Stroock, D. W.. Large Deviation (New York: Academic Press, 1989).Google Scholar
7Elworthy, K. D.. Stochastic Differential Equations on Manifold, London Mathematical Society Lecture Notes Series 70 (Cambridge: Cambridge University Press, 1982).CrossRefGoogle Scholar
8Elworthy, K. D. and Truman, A.. Classical mechanics, the diffusion (heat) equations and the Schrödinger equation on a Riemannian manifold. J. Math. Phys. 22 (1981), 2144–2166.CrossRefGoogle Scholar
9Elworthy, K. D. and Truman, A.. The diffusion equation and classical mechanics: an elementary formula. In Stochastic Processes in Quantum Physics, eds Albeverio, S.et al., Lecture Notes in Physics 173, 136–146 (Berlin: Springer, 1982).CrossRefGoogle Scholar
10Fife, P. C.. Mathematical Aspects of Reacting and Diffusion Systems, Lecture Notes in Biomathematics 28 (Berlin: Springer, 1979).CrossRefGoogle Scholar
11Fife, P. C. and McLeod, J. B.. The approach of solutions of nonlinear diffusion equations to travelling wave solutions. Arch. Rational Mech. Anal. 65 (1977), 335–361.CrossRefGoogle Scholar
12Fisher, R. A.. The wave of advance of advantageous genes. Ann. Eugenics 7 (1937), 353–369.CrossRefGoogle Scholar
13Fitzgibbon, W. E. (III) and Walker, H. F. (eds). Nonlinear Diffusion, Research Notes in Mathematics 14 (London: Pitman, 1977).Google Scholar
14Freidlin, M. I.. Propagation of a concentration wave in the presence of random motion association with the growth of a substance. Soviet Math. Dokl. 20 (1979), 503–507.Google Scholar
15Freidlin, M. I.. On wave fronts propagation in multicomponent media. Trans. Amer. Math. Soc. 276 (1983), 181–191.CrossRefGoogle Scholar
16Freidlin, M. I.. Functional Integration and Partial Differential Equations (Princeton: Princeton University Press, 1985).Google Scholar
17Freidlin, M. I.. Coupled reaction–diffusion equations. Ann. Probab. 19 (1991), 29–57.CrossRefGoogle Scholar
18Freidlin, M. I.. Semi-linear PDE's and limit theorem for large deviations. In Ecole d'Eté Probabilités de Saint-Flour XX-1990, ed. P. Hennequin, L., Lecture Notes in Mathematics 1527, 1–109. (Berlin: Springer, 1992).CrossRefGoogle Scholar
19Frobenius, G.. Uber Matrizen aus Positiven Elementen. Deutch, S.-B.. Akad. Wiss. Berlin. Math-Nat. Kl. (1908), 471–476.Google Scholar
20Frobenius, G.. Uber Matrizen aus Positiven Elementen. Deutch, S.-B.. Akad. Wiss. Berlin. Math-Nat. Kl. (1909), 514–518.Google Scholar
21Frobenius, G.. Uber Matrizen nicht Negative Elementen. Deutch, S.-B.. Akad. Wiss. Berlin. Math-Nat. Kl. (1912), 456–477.Google Scholar
22Gantmacher, F. R.. The Theory of Matrices, Vol. II (New York: Chelsea, 1974).Google Scholar
23Gartner, J.. Location of wave fronts for the multi-dimensional K–P–P equation and Brownian first exist densities. Math. Nachr. 105 (1982), 317–351.CrossRefGoogle Scholar
24Kolmogoroff, A., Petrovsky, I. and Piscounoff, N.. Etude de l'equation de la diffusion avec croissance de la matière et son application à une problème biologique. Moscow Uni. Math. Bull. 1 (1937), 1–25.Google Scholar
25Lakshmikantham, V. and Leela, S.. Differential and Integral Inequalities, Theory and Applications, Vol. 1, Ordinary Differential Equations (New York: Academic Press, 1969).Google Scholar
26Larson, D. A.. Transient bounds and time asymptotic behavior of solutions of nonlinear equations of Fisher type. SIAM J. Appl. Math. 34 (1978), 93–103.CrossRefGoogle Scholar
27Leandre, R.. A simple proof for a large deviation theorem (Proceedings of Bulefeld Conference on White Noise, 1989).Google Scholar
28Luther, R.-L.. Raumliche fortplanzung chemischer reaktionen. Z. Elektrochemie Angew, Physikalische Chemie 12(32) (1906), 506–600; English translation: Arnold, R., Schowalter, K. and Tyson, J. J.. Propagation of chemical reaction in space. J. Chem. Education (1988).Google Scholar
29Manoranjan, V. S. and Mitchell, A. R.. A numerical study of the Belousov–Zhabotinskii reaction using Galerkin finite elements methods. J. Math. Biol. 16 (1983), 251–260.CrossRefGoogle Scholar
30McKean, H. P.. Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 28 (1975), 323–331.CrossRefGoogle Scholar
31McKean, H. P.. A correction to: Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 29 (1975), 553–554.CrossRefGoogle Scholar
32Molchanov, S. A.. Diffusion processes and Riemannian geometry. Mat. Uspekhi Nauk 30 (1975), 3–59 (translated in Russian Math. Surveys 30 (1975), 1–63).Google Scholar
33Mollison, D.. Spatial contact models for ecological and epidemic spread. J. Roy. Statist. Soc. Ser. B 39(1977), 283–326.Google Scholar
34Murray, J. D.. Mathematical Biology (Berlin: Springer, 1989).CrossRefGoogle Scholar
35Pinsky, M. A.. Stochastic integral representations of multiplicative operator functionals of a Weiner process. Trans. Amer. Math. Soc. 167 (1972), 89–104.CrossRefGoogle Scholar
36Pinsky, M. A.. Multiplicative operator functionals and their asymptotic properties. Adv. Probab. 3 (1974), 1–100.Google Scholar
37Schilder, M.. Some asymptotic formula for Wiener integrals. Trans. Amer. Math. Soc. 125 (1966), 63–85.CrossRefGoogle Scholar
38Segal, L. A. (ed.). Mathematical Models in Molecular and Cellular Biology (Cambridge: Cambridge University Press, 1980).Google Scholar
39Smoller, J.. Shock Waves and Reaction–Diffusion Equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
40Stroock, D. W.. On certain systems of parabolic equations. Comm. Pure Appl. Math. 23 (1970), 447–457.CrossRefGoogle Scholar
41Stroock, D. W.. An Introduction to the Theory of Large Deviations (Berlin: Springer, 1984).CrossRefGoogle Scholar
42Varadhan, S. R. S.. Asymptotic probabilities and differential equations. Comm. Pure Appl. Math. 19 (1966), 268–286.CrossRefGoogle Scholar
43Varadhan, S. R. S.. Diffusion process in a small time interval. Comm. Pure Appl. Math. 20 (1967), 659–685.CrossRefGoogle Scholar
44Varadhan, S. R. S.. On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20 (1967), 431–455.CrossRefGoogle Scholar
45Volpert, A. I. and Volpert, V. A.. Applications of the rotation theory to vector fields to the study of the wave solutions of parabolic equations. Trans. Moscow Math. Soc. (1990), 59–108.Google Scholar
46Wentzell, A. D. and Freidlin, M. I.. On small random perturbations of dynamical systems. Upsekhi Mat. Nauk 28(1) (1970), 3–55.Google Scholar
47Winfree, A. T.. The Geometry of Biological Time (Berlin: Springer, 1980).CrossRefGoogle Scholar
48Zhao, H. Z. and Elworthy, K. D.. The travelling wave solutions of scalar generalized KPP equations via classical mechanics and stochastic approaches. In Stochastic and Quantum Mechanics, eds Davies, I. and Truman, A., 298–316 (Singapore: World Scientific Press, 1992).Google Scholar