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In this work, we mainly consider the Cauchy problem for the reversespace-time nonlocal Hirota equation with the initial data rapidly decaying inthe solitonless sector. Start from the Lax pair, we first construct the basisRiemann-Hilbert problem for the reverse space-time nonlocal Hirota equation.Furthermore, using the approach of Deift-Zhou nonlinear steepest descent, theexplicit long-time asymptotics for the reverse space-time nonlocal Hirota isderived. For the reverse space-time nonlocal Hirota equation, since thesymmetries of its scattering matrix are different with the local Hirotaequation, the $\vartheta(\lambda_{i})(i=0, 1)$ would like to be imaginary,which results in the $\delta_{\lambda_{i}}^{0}$ contains an increasing$t^{\frac{\pm Im\vartheta(\lambda_{i})}{2}}$, and then the asymptotic behaviorfor nonlocal Hirota equation becomes differently.

Long-time asymptotics for the reverse space-time nonlocal Hirota equation with decaying initial value problem: Without solitons