Stability of nonlinear waves for hyperbolic conservation laws will viscosity

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Our aim in this research is to investigate the asymptoic behavior of time-global solutions for one dimensional compressible flow with viscosity due to the Newton viscosity or the friction. The system is written by the hyperbolic conservation laws with viscosity, which has the nonlinear waves like viscous shock wave, rare faction wave, diffusion wave and the wave corresponding to contact discontinuity.For the compressible Navier-Stokes equation the global stability of strong rare faction wave is shown, whose method is applied to the Jin-Xin relaxation model for p-system. Also, in the inflow problem on half-line the solution is shown to tend the superposition of viscous shock wave and boundary layer under some conditions, in which case the problem was open.On the other hand, the p-system with friction is modeled by the compressible flow in porous media. The solution was shown by Hsiao-Liu to approach to the solution of the corresponding parabolic system due to the Darcy law. Through the precise consideration of the approach we have reached to the fact that the damped wave equation of second order is closely related to the corresponding heat equation in one and three dimensional space, which is applied to show the existence of time-global solution or the blow-up of solution in a finite time for the semilinear damped wave equation. The critical exponent is same as that in the semilinear heat equation, which is reasonably understood by the fact obtained. It is also seen in the abstract setting. So, our result may give some suggestions in the investigation on the damped wave equation and related problem.