1998/09/18 by Th. Gallay, Gallay, Th., G. Raugel +1
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #FOS: Physical sciences #Nonlinear Partial Differential Equations #Pattern Formation and Solitons (nlin.PS) #Stability and Controllability of Differential Equations #nlin.PS #patt-sol
paper · pdf · doi:10.48550/arxiv.patt-sol/9809007
20 pages, plain TeX
arxiv created 1998/09/18 · openalex publication_date 1998/09/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the damped hyperbolic equation in one space dimension εutt + ut = uxx + F(u), where ε is 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 assumptions, our equation has a continuous family of monotone propagating fronts (or travelling waves) indexed by the speed parameter c ≥ c_*. Using energy estimates, we first show that the travelling waves are locally stable with respect to perturbations in a weighted Sobolev space. Then, under additional assumptions on the non-linearity, we obtain global stability results using a suitable version of the hyperbolic Maximum Principle. Finally, in the critical case c = c_*, we use self-similar variables to compute the exact asymptotic behavior of the perturbations as t → +∞. In particular, setting ε= 0, we recover several stability results for the travelling waves of the corresponding parabolic equation.