Global well-posedness and time-decay estimates for compressible Navier-Stokes equations with reaction diffusion

We consider the full compressible Navier-Stokes equations with reaction diffusion. A global existence and uniqueness result of the strong solution is established for the Cauchy problem when the initial data is in a neighborhood of a trivially stationary solution. The appearance of the difference between energy gained and energy lost due to the reaction is a new feature for the flow and brings new difficulties. To handle these, we construct a new linearized system in terms of a combination of the solutions. Moreover, some optimal timedecay estimates of the solutions are derived when the initial perturbation is additionally bounded in L1. It is worth noticing that there is no decay loss for the highest-order spatial derivatives of the solution so that the long time behavior for the hyperbolic-parabolic system is exactly the same as that for the heat equation. As a byproduct, the above time-decay estimate at the highest order is also valid for compressible Navier-Stokes equations. The proof is accomplished by virtue of Fourier theory and a new observation for cancellation of a low-medium-frequency quantity.