Analysis of non-isentropic compressible Euler equations with relaxation

This paper is contributed to the study of the one-dimensional non-isentropic compressible Euler equations with relaxation. It is shown that classical solutions do not exist globally-in-time under general conditions on initial data. Indeed, finite-time blowup occurs in a quantity related to the first moment. On the other hand, when the initial datum is sufficiently close to a constant equilibrium state, it is shown that the equations possess a unique global-in-time classical solution, and the solution converges to the equilibrium state in the long-time run. When the domain is finite, the convergence rate is shown to be exponential, due to boundary effects.