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.
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1.
Let u(x, 0)=u 0(x) satisfy 0≦u 0≦1. Let \(a\_ = \mathop {\lim \sup u0}\limits_{x \to - \infty } {\text{(}}x{\text{), }}\mathop {\lim \inf u0}\limits_{x \to \infty } {\text{(}}x{\text{)}}\). 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.
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2.
Suppose \(\int\limits_{\text{0}}^{\text{1}} {f{\text{(}}u{\text{)}}du} > {\text{0}}\). 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.
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3.
Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges.
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This research was sponsored in part by the United States Army under Contract No. DAAG29-75-C0024, in part by the National Science Foundation under Grant MPS-74-06835-A01, and in part by a Science Research Council British Research Council British Research Fellowship.
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Fife, P.C., McLeod, J.B. The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rational Mech. Anal. 65, 335–361 (1977). https://doi.org/10.1007/BF00250432
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DOI: https://doi.org/10.1007/BF00250432