Th. Gallay and G. Raugel (Paris XI)
Stability of Propagating Fronts in Damped Hyperbolic Equations
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ABSTRACT. 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.