Asymptotic behavior of multidimensional scalar viscous shock fronts

Making use of detailed pointwise Green's function bounds obtained in a prev ious work for the linearized equations about the wave, we give a straightfo rward derivation of the (nonlinear) L-P-asymptotic behavior of;a scalar (pl anar) viscous shock front under perturbations in L-1 boolean AND L-infinity with first moment in the normal direction to the front, in all dimensions d greater than or equal to 2. For dimension d greater than or equal to 3, w e establish sharp LP decay rates by a much simpler argument using only L-P information on the Green's function, for perturbations merely in L-1 boolea n AND L-infinity. These results simplify and greatly extend previous result s of Goodman-Miller and Goodman, respectively which were obtained under ass umptions of weak shock strength and artificial (identity) viscosity, and, i n the case of asymptotic behavior, exponential decay of perturbations in th e direction normal to the shock front. For perturbations localized as (1+\x (1)\)(-1) in the normal direction, but not possessing a first moment, we gi ve a refined picture of the linearized L-P-asymptotic behavior different fr om the near-field approximation of Goodman and Miller.