A Phase Plane Discussion of Convergence to Travelling Fronts for Nonlinear Diffusion.
Abstract
The paper is concerned with the asymptotic behavior as t approaches infinity of solutions u(x,t) of the equation u sub t-u sub xx-f(u)=0, x an element, from -infinity to infinity, 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(-infinity)=0, U(infinity)=1, 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 sub x as the dependent variable, and apply ideas of sub- and super-solutions to the diffusion equation for p. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1979
- Accession Number
- ADA077131
Entities
People
- J. B. Mcleod
- Paul C. Fife
Organizations
- University of Wisconsin–Madison