Asymptotic decay to relaxation shock fronts in two dimensions (Q2784205)

The author proves the time-asymptotic stability of planar shock fronts for certain relaxation systems in two spatial dimensions. The initial perturbation is assumed to be sufficiently small, but the mass carried by the perturbation is not generally finite. The subcharacteristic condition is supposed to be satisfied and it plays an essential role in the stability analysis. The author establishes that the perturbed solution converges to the shifted planar shock front solution as time \(t\to\infty\), and the asymptotic phase shift is governed by a similarity solution of the heat equation. The asymptotic decay rate of the shock front is proved to be \(t^{-1/4}\) in \(L^\infty({\mathbb R}^2)\).