Long time behavior of solutions to the compressible MHD system in multi-dimensions

In this paper, we consider the initial value problem for the compressible MHD system in multi-dimensions. Under suitable conditions on the initial value, the global existence and long time behavior of classical solutions are established. Moreover, we also show that the solution can be approximated by the linear solution as time tends to infinity. The proof is based on the decay properties of solution operator to the compressible MHD system, which may be given in terms of the heat kernel and the solution operator to a linear wave equation that is corresponding to the density function. The decay properties of solution operators to the compressible MHD system may be derived from the pointwise estimate of solution operator to the linear wave equation. Finally, asymptotic profile to the linear system is also established.