Optimal temporal decay rates of solutions for combustion of compressible fluids

Abstract

This paper investigates the temporal decay rates of solutions to the Cauchy problem of a model, which describes the combustion of the compressible fluid. Suppose that the initial data is a small perturbation near the equilibrium state \((\rho _\infty , 0,\theta _\infty ,\zeta )\), where \(\rho _\infty >0\), \(\theta _\infty <\theta _I\) (the ignition temperature), and \(0< \zeta \leqslant 1\), we first establish the global-in-time existence of strong solutions via a standard continuity argument. With the additional \(L^1\)-integrability of the initial perturbation, we then employ the Fourier theory and the cancellation mechanism of low-medium frequent part to derive the optimal temporal decay rates of all-order derivatives of strong solutions. Our work is a natural continuation of previous result in the case of \(\theta _\infty >\theta _I\) discussed in Wang and Wen (Sci China Math 65:1199–1228 (2022).