2000/12/19 by Elaine Crooks, E. C. M. Crooks, Crooks, E. C. M.
Computer Science · Mathematics · Medicine · #35B35 #35B40 #35K40 #35K45 #35K55 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation #math.AP #msc:35B35 #msc:35B40 #msc:35K40 #msc:35K45 #msc:35K55
paper · pdf · doi:10.48550/arxiv.math/0012181
23 pages. To appear in Topological Methods in Nonlinear Analysis
arxiv created 2000/12/19 · openalex publication_date 2000/12/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are concerned with the asymptotic behaviour of classical solutions of systems of the form ut = Auxx + f(u, ux), x in R, t>0, u(x,t) a vector in RN, with u(x,0)= U(x), where A is a positive-definite diagonal matrix and f is a 'bistable' nonlinearity satisfying conditions which guarantee the existence of a comparison principle. Suppose that there is a travelling-front solution w with velocity c, that connects two stable equilibria of f. We show that if U is bounded, uniformly continuously differentiable and such that w(x) - U(x) is small when the modulus of x is large, then there exists y in R such that u(., t) converges to w(.+y-ct) in the C1 norm at an exponential rate as t tends to infinity. Our approach extends an idea developed by Roquejoffre, Terman and Volpert in the convectionless case, where f is independent of ux.