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Diffusive waves in inhomogeneous media

Published online by Cambridge University Press:  20 January 2009

Paul C. Fife
Paul C. Fife
Affiliation:
Heriot–Watt University, Edinburgh, Scotland University of Utah, Salt Lake City, Utah, U.S.A.
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When the function f(u) is of “bistable type’, i.e. has two zeros h̲ and h+ at which f' is negative and (for simplicity) has only one other zero between them, then the constant functions u = h± are L∞-stable solutions of the nonlinear diffusion equation

In addition, there are travelling wave solutions u+(x, t) and u̲(x, t) which, if

connect h+ to h̲ in the sense that

the convergence being uniform on bounded x-intervals. These solutions are of the form

where U(z) is a monotone function (the wave's profile), U(±∞) = h±, and the velocity c is a specific positive number depending on the function f.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 2 , June 1989 , pp. 291 - 315
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCE

1.Fife, P. C. and Hsiao, L., The generation and propagation of internal layers, Nonlinear Anal., TMA, 12 (1988), 19–41.CrossRefGoogle Scholar
2.Fife, P. C. and McLeod, J. B., The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335–361.CrossRefGoogle Scholar
3.Fife, P. C., Long time behavior of solutions of bistable nonlinear diffusion equations, Arch. Rat. Mech. Anal. 70 (1979), 31–46.CrossRefGoogle Scholar
4.Fife, P. C., and Peletier, L. A., Clines induced by variable selection and migration, Proc. Roy. Soc. London B214 (1981), 99–123.Google Scholar
5.Fife, P. C., and Peletier, L. A. Unidirectional barriers to the propagation of fronts, in preparation.Google Scholar
6.Fisher, R. A., The advance of advantageous genes, Ann. of Eugenics 7 (1937), 355–369.CrossRefGoogle Scholar