Initial–boundary value problem for p-system with damping in half space

In this paper, the initial–boundary value problem for p-system with damping in half space is studied with a posed mix boundary condition. The fundamental solution (Green’s function for Cauchy problem) is constructed with a sharp decaying structure and a precise capture of the singularity by a “singularity removal” process. Later, the symbols of both fundamental solution and Green’s function (for initial–boundary value problem) are obtained with transformed variable s and spatial variable x while in this level the Green’s function could be decomposed into fundamental solution and boundary operator. Thus, with the study of boundary operator and previous precise information about the fundamental solution, the pointwise structure as well as the singular structure of the Green’s function is established. Finally, the nonlinear stability is obtained by the Green’s function and a priori estimate from energy method.