The approach of solutions of nonlinear diffusion equations to travelling front solutions

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Abstract

The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut-uxx-∞;(u)=O, x∈(-∞, ∞), in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given. 1. Let u(x, 0)=u0(x) satisfy 0≦u0≦1. Let {Mathematical expression}. 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. 2. Suppose {Mathematical expression}. If a- and a+ are not too far from 0, but u0 exceeds a 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. © 1977 Springer-Verlag.

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Fife, P. C., & McLeod, J. B. (1977). The approach of solutions of nonlinear diffusion equations to travelling front solutions. Archive for Rational Mechanics and Analysis, 65(4), 335–361. https://doi.org/10.1007/BF00250432

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