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The approach of solutions of nonlinear diffusion equations to travelling wave solutions (1975)
| Citations: | 235 - 5 self |
BibTeX
@MISC{Fife75theapproach,
author = {Paul C. Fife and J. B. Mcleod},
title = {The approach of solutions of nonlinear diffusion equations to travelling wave solutions},
year = {1975}
}
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Abstract
The paper is concerned with the asymptotic behavior as t---, oo of solutions u(x, t) of the equation in the case ut-uxx-f(u)=O, xe(- ~, oo), f(0) =f(1) =0, f'(0)<0, f'(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-oo)=0, U(oo)=l, exists. The following types of global stability results for fronts and various combinations of them will be given. 1. Let u(x,O)=uo(X) satisfy 0<u0<l. Let a _ = lim sup Uo(X), a + = lira infuo(X). Then u approaches a translate of U uniformly in x and exponentially in time, if a is not too far from 0, and a+ not too far from 1. 1 2. Suppose ~f(u)du>O. Ifa and a+ are not too far from 0, but u o exceeds a o certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts. 3. Under certain circumstances, u approaches a "stacked " combination of wave fronts, with differing ranges. 1.
