Asymptotic stability of the stationary solution to the three-dimensional model of compressible reactive fluid

Abstract

In this paper, we consider the asymptotic behavior of solutions to the Cauchy problem for the three-dimensional model of compressible reactive fluid, which can be described by a compressible Navier–Stokes type system with potential external force. First, the existence of the stationary solution is shown in the case that the external force is small enough. Next, making use of the energy method, we prove that the stationary solution is time-asymptotically stable provided that the external force and the initial perturbation are sufficiently small. Finally, we obtain the time-decay rate of the solution toward the stationary solution by combining the $L^p-L^q$ estimates for the corresponding linear problem and the energy estimates for the nonlinear system.