Stability of Propagating Fronts in Damped Hyperbolic Equations
We consider the damped hyperbolic equation ε u t t + u t = u x x + F ( u ) , x ∈ R, t ≥ 0 , u ∈ R, where ϵ is a positive, not necessarily small parameter. We assume that F(0)=F(l)=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 → +∞ +oo. In particular, setting ϵ=0, we recover several stability results for the travelling waves of the corresponding parabolic equation.