Hardy type inequality and application to the stability of degenerate stationary waves

Abstract

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t − α / 4 as t → ∞ , provided that the initial perturbation is in the weighted space L α 2 = L 2 ( R + ; ( 1 + x ) α ) for α < α c ( q ) : = 3 + 2 / q , where q is the degeneracy exponent. This restriction on α is best possible in the sense that the corresponding linearized operator cannot be dissipative in L α 2 for α > α c ( q ) . Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant.