Scaling Variables and Stability of Hyperbolic Fronts

Details Scaling Variables and Stability of Hyperbolic Fronts Scaling Variables and Stability of Hyperbolic Fronts

1998-12-18

arxiv.org/abs/patt-sol/9812007v1

Nonlinear Sciences

Pattern Formation and Solitons

38 pages, plain TeX

Scientific paper

We consider the damped hyperbolic equation (1) \epsilon u_{tt} + u_t = u_{xx} + F(u), x \in R, t \ge 0, where \epsilon 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 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 \ge 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 \to +\infty and approach a universal self-similar profile, which is independent of \epsilon, F and of the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting \epsilon = 0 in Eq.(1). The proof of our results relies on careful energy estimates for the equation (1) rewritten in self-similar variables x/\sqrt{t}, \log t.

Affiliated with

Also associated with

No associations

Rating

Scaling Variables and Stability of Hyperbolic Fronts does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Scaling Variables and Stability of Hyperbolic Fronts, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Scaling Variables and Stability of Hyperbolic Fronts will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-348019