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　(ρ∞,0,θ∞,ζ),　where　ρ∞＞0,　θ∞＜θI　(the　ignition　temperature),　and　0＜ζ⩽1,　we　first　establish　the　global-in-time　existence　of　strong　solutions　via　a　standard　continuity　argument.　With　the　additional　L1-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　θ∞＞θI　discussed　in　Wang　and　Wen　(Sci　China　Math　65:1199–1228　(2022).　©　The　Author(s),　under　exclusive　licence　to　Springer　Nature　Switzerland　AG　2024.