Study on the initial value problem for quasilinear hyperbolic-elliptic coupled systems

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We study the initial value problem for a class of hyperbollic-elliptic coupled systems including the equation of a radating gas. For the simplest model system of a radiating gas :1.We give sufficient conditions for the non-existence and the existence of classical global-solutions.2.We prove the global existence and asymptotic decay of smooth solutions in Sobolev spaces. The solution approaches the diffusion wave which is defined in terms of the self-similar solution of the viscous Burgers equation.3.We prove the asymptotic stability of rarefaction waves which are defined in terms of the centered rarefaction wave of the inviscid Brugers equation.4.We show the existence of traveling wave solutions of shock profile. These shock waves have a discontinuity only when the shock strength is greater than a critical value. Moreover, we prove the asymptotic stability of smooth shock waves.5.We show the glogal existence of weak solutions to the Riemann problem. The jump contained in the weak solution decays exponentially, and the solution approaches the corresponding smooth shock wave as time tends to infinity.6.We give a mathematical definition of the entropy function and prove the equivalence of the existence of an entropy function and the symmetrization of the system. Next, we formulate the stability condition, and under that condition we show the global existence and asymptotic decay of smooth solutions in Sobolev spaces. Furthermore, we observe that the solution is time-asymptotically approximated by the solution to the eorresponding hyperbolic-parabolic coupled system. These results are applicable to the equation of a radiating gas.