Dirichlet-neumann Kernel for Hyperbolic-dissipative System in Half-space

The purpose of the present paper is to initiate a systematic study of the relation of the boundary values for hyperbolic-dissipative systems of partial differential equations. We introduce a general framework for explicitly deriving the boundary kernel for the Dirichlet-Neumann map. We first use the Laplace and Fourier transforms, and the stability consideration to derive the Master Relationship, the Dirichlet-Neumann relation in the transformed variables. New idea of Fourier-Laplace path and algebraic considerations are introduced for the explicit inversion of Fourier-Laplace transforms. We illustrate the basic ideas by carrying out the framework to models in the gas dynamics and the dissipative wave equations.