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 $s({\text{eps}}^{\mathrm{\prime }},k)>0$, where ${\text{eps}}^{\mathrm{\prime }}=\frac{T_{min}}{T_{max}}\ll 1$ is a small temperature ratio and $k\gg 1$ the transversal wrinkling wavenumber of perturbations. We rigorously recover in some particular limit $({\text{eps}}^{\mathrm{\prime }},k)\to (0,+\mathrm{\infty })$ a dispersion relation $s({\text{eps}}^{\mathrm{\prime }},k)\sim {\gamma }_0{k}^{\alpha }$ previously computed heuristically and numerically in some physical models of Inertial Confinement Fusion.