Damped Euler-Poisson equations and nonlinear diffusion waves

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Compressible flow through porous media with a dissipative external force field is usually described as a hyperbolic p-system with damping, a kind of Euler system of partial differential equations. The damping effect makes the system behave as a set of nonlinear diffusion equations, and the solutions possessing diffusion characters are known as nonlinear diffusion waves. Similar phenomena also occur in the hydrodynamic system for the models of semiconductor devices (Euler-Poisson equations). To study if the solutions of the original complex system behave as those diffusion waves and how to converge to the diffusion waves in the subsonic case, and to investigate the existence of the weak entropy solutions and their large-time behavior in the transonic case, the supersonic case, the case with vacuum, are very significant and challenging from both mathematical and physical points of view, and have been the hot research spots in applied partial differential equations