Linear relaxation to planar Travelling Waves in Inertial Confinement Fusion

We study linear stability of planar travelling waves for a scalar reaction-diffusion equation with non-linear anisotropic diffusion. The mathematical model is derived from the full thermo-hydrodynamical model describing the process of Inertial Confinement Fusion. We show that solutions of the Cauchy problem with physically relevant initial data become planar exponentially fast with rate $s(\eps',k)>0$, where $\eps'=\frac{T_{min}}{T_{max}}\ll 1$ is a small temperature ratio and $k\gg 1$ the transversal wrinkling wavenumber of perturbations. We rigorously recover in some particular limit $(\eps',k)\rightarrow (0,+\infty)$ a dispersion relation $s(\eps',k)\sim \gamma_0 k^{\alpha}$ previously computed heuristically and numerically in some physical models of Inertial Confinement Fusion.