The approach of solutions of nonlinear diffusion equations to travelling front solutions View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

1977-12

AUTHORS

Paul C. Fife, J. B. McLeod

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.Let u(x, 0)=u0(x) satisfy 0≦u0≦1. Let . 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.Suppose . 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.Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges. Let u(x, 0)=u0(x) satisfy 0≦u0≦1. Let . 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. Suppose . 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. Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges. More... »

PAGES

335-361

References to SciGraph publications

  • 1975. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation in PARTIAL DIFFERENTIAL EQUATIONS AND RELATED TOPICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf00250432

    DOI

    http://dx.doi.org/10.1007/bf00250432

    DIMENSIONS

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