Stability of Propagating Fronts in Damped Hyperbolic Equations

We consider the damped hyperbolic equation in one space dimension $\epsilon u_{tt} + u_t = u_{xx} + F(u)$, 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 assumptions, our equation has a continuous family of monotone propagating fronts (or travelling waves) indexed by the speed parameter $c \ge c_*$. 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 $t \to +\infty$. In particular, setting $\epsilon = 0$, we recover several stability results for the travelling waves of the corresponding parabolic equation.