1998/12/18 by Th. Gallay, Gallay, Th., G. Raugel +1 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Computational Fluid Dynamics and Aerodynamics #FOS: Physical sciences #Numerical methods for differential equations #Pattern Formation and Solitons (nlin.PS) #Quantum chaos and dynamical systems #nlin.PS #patt-sol
paper · pdf · doi:10.48550/arxiv.patt-sol/9812007
38 pages, plain TeX
arxiv created 1998/12/18 · openalex publication_date 1998/12/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the damped hyperbolic equation (1) εutt + ut = uxx + F(u), x ∈ R, t ≥ 0, where εis a positive, not necessarily small parameter. We assume that F(0) = F(1) = 0 and that F is concave on the interval [0,1]. Under these hypotheses, Eq.(1) has a family of monotone travelling wave solutions (or propagating fronts) connecting the equilibria u=0 and u=1. This family is indexed by a parameter c ≥ c_* related to the speed of the front. In the critical case c=c_*, we prove that the travelling wave is asymptotically stable with respect to perturbations in a weighted Sobolev space. In addition, we show that the perturbations decay to zero like t-3/2 as t → +∞ and approach a universal self-similar profile, which is independent of ε, F and of the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting ε= 0 in Eq.(1). The proof of our results relies on careful energy estimates for the equation (1) rewritten in self-similar variables x/√(t), log t.