Stability of nonlinear waves in viscous conservation system together with diffusion phenomena of solutions of damped wave equation

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The system of conservation laws has the shock wave, rarefaction wave and contact discontinuity as nonlinear waves. In real physics, it may become the system of viscous conservation laws by some viscous effect, which yields the viscous shock wave, rarefaction wave and viscous contact wave with diffusion wave. Our aim of this research is to observe the stability of the waves.In our research the viscous effect is by the usual Newton viscosity or friction in porous media flow. The flow approaches to the solution of corresponding parabolic system by Darcy's law, which implies that the damped wave equation behaves as the corresponding diffusion equation as time tends to infinity, what we call the diffusion phenomena. The observation of this phenomena is another aim of this research. The stability of viscous contact wave in the viscous conservation laws with Newton's viscosity has been mainly developed by the investigator. The diffusion phenomena of solutions to the damped wave equation has been investigated by the head investigator, based on the fact that the solution of the Cauchy problem far the linear damped wave equation is decomposed to the sum of the wave part exponentially decaying and the diffusion part For the corresponding diffusion equation, rather precise results are obtained thanks to the smoothing effects and the maximum principle, but these key properties do not hold for the wave equation, and further studies are necessary.