Cauchy problem for a model system of the radiating gas: Weak solutions with a jump and classical solutions

This paper deals with the global existence and the time asymptotic state of solutions to the initial value problems for the system derived from approximating a one-dimensional model of a radiating gas. When the spatial derivative of the initial data is larger than a certain negative critical value, a unique solution exists globally in time. But if it is smaller than another negative critical value, the spatial derivative of the corresponding solution blows up in a finite time. Thus it is natural to think about weak solutions in a suitable sense. As a prototype of weak solutions, we consider the Cauchy problem with the Riemann initial data of which the left state is larger than the right state. This condition ensures the existence of the corresponding traveling wave, connecting the left state and the right state asymptotically. This Riemann problem admits a global weak solution, provided that the magnitude of the initial discontinuity is smaller than 1/2. Although the solution has a discontinuity, we have the uniqueness of a solution in weak sense by imposing the entropy condition. Furthermore, the magnitude of the discontinuity contained in the solution decays to zero with an exponential rate as the time t goes to infinity. Also, the solution approaches the corresponding traveling wave with the rate t^-1/4 uniformly. The first result is obtained by the maximal principles. To show the second result, we have used an energy method with some estimates, which are also obtained through maximal principles.