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A phase plane discussion of convergence to travelling fronts for nonlinear diffusion

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Abstract

The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x,t) of the equation \(u_t - u_{xx} - f\left( u \right) = 0, x \in \left( { - \infty ,\infty } \right),\) in the case f(0)=f(1)=0, with f(u) non-positive for u(>0) sufficiently close to zero and f(u) non-negative for u(<1) sufficiently close to 1. This guarantees the uniqueness (but not the existence) of a travelling front solution u;U(x−ct), U(−∞);0, U(∞);, and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front, if it exists, and to a “stacked” combination of travelling fronts if it does not. The approach is to use the monotonicity to take u and t as independent variables and p = u x as the dependent variable, and to apply ideas of sub- and super-solutions to the diffusion equation for p.

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References

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson

    Paul C. Fife & J. B. McLeod

  2. Wadham College, Oxford

    Paul C. Fife & J. B. McLeod

Authors
  1. Paul C. Fife
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  2. J. B. McLeod
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Additional information

This research was sponsored by the United States Army under Contract No. DAAG29-75-C-0024.

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Fife, P.C., McLeod, J.B. A phase plane discussion of convergence to travelling fronts for nonlinear diffusion. Arch. Rational Mech. Anal. 75, 281–314 (1981). https://doi.org/10.1007/BF00256381

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  • DOI: https://doi.org/10.1007/BF00256381

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