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Hyperbolic travelling fronts

Published online by Cambridge University Press:  20 January 2009

K. P. Hadeler
K. P. Hadeler
Affiliation:
Lehrstuhl für BiomathematikUniversität Tübingenauf der Morgenstelle 10D-7400 TübingenWest Germany
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Since the work of R. A. Fisher [2] and Kolmogorov, Petrovskij and Piskunov [7] (see [6] for further references) the problem of travelling fronts in reaction–diffusion equations has been extensively studied. For the equation

with F(0) = F(1) = 0 a travelling front is a solution

where the function of one variable φ is decreasing and satisfies φ(−∞) = 1, φ(+∞) = 0. The function φ describes the shape of the front and the constant c is the speed of propagation. There are two main types of the problem. In the all-positive case, where the function F satisfies

there is a half-line [c0, ∞), c0>0, of speeds. For each c∈[c0, ∞) there is, up to translation, a unique travelling front. Fronts for different c can be distinguished by the rate of decay towards +∞. In the threshold case, where F has the property, for some λ∈(0, 1),

there is a unique speed c0 with a travelling front, which is unique up to translation. In this case the sign of c0 is determined by the sign of the integral

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , Issue 1 , February 1988 , pp. 89 - 97
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

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