Initial-boundary value problem of Euler equations with damping in R+n

The asymptotic wave behavior of solution for the Euler equations with damping in R+n is investigated around a given constant equilibrium in this paper. Three classical boundary condition cases are considered here. New approaches and techniques are introduced to deal with the multi-dimensional case for the system. For the linearized problem, by comparing symbols in the transformed tangential-spatial and time space, we show that its Green's function can be described in terms of the fundamental solution for the Cauchy problem and reflected fundamental solution coupled with a boundary operator. Our approaches help us to simplify the Green's function to the essential part and benefit the follow-up nonlinear analysis. For the nonlinear problem, we prove the pointwise decaying rate with the help of the Duhamel's principle.