Blow up and global existence of solutions for a model system of the radiating gas

In this paper, we are concerned with the Cauchy problem for a model system of the radiating gas in multi-dimensional space. We first show the blow up of classical solutions to the nonlinear Cauchy problem by employing the characteristic method. On the basis of this blow up result, when the initial data u0(x)u0(x) satisfies the smallness conditions on ‖∇u0‖Hs‖∇u0‖Hs, but not on ‖u0‖L2‖u0‖L2, the global existence of solutions to the Cauchy problem is established. Our approach is based on a low frequency–high frequency decomposition and the classical energy method. Furthermore, by taking a time–frequency decomposition, we prove the optimal time-decay rates of the solutions.