<p>An asymptotic analysis was performed for direct detonation initiation in unsteady flow at elevated pressure, which extends our previous work for planar wave front. The reaction zone temperature-gradient equation was established and then solved asymptotically by assuming large activation energy and a one-step irreversible reaction model. The Noble-Abel and van der Waals gas models were adopted to study the impact of real-gas (RG) behavior. While the attractive intermolecular force makes initiation more difficult by increasing the critical shock decay time, the repulsive force promotes initiation. The effective range of divergence time is extended/limited by the attraction/repulsion force. Except for mixtures with large heat capacity ratio of the perfect gas (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\gamma ^{\circ }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>γ</mi> <mo>∘</mo> </msup> </math></EquationSource> </InlineEquation>) and small reduced activation energy with respect to the initial state (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>) and satisfying the blast wave model of Korobeinikov, the impact of the divergence time on the critical decay time is insignificant because the critical state is reached at large blast wave radius, indicating that the curvature effect is limited. The RG effects are enhanced when increasing the reduced activation energy or/and decreasing the perfect gas heat capacity ratio. For the ease of applying the present findings, the asymptotic solutions were firstly validated with quasi-unsteady simulation using detailed reaction models. It was then further simplified to obtain fully analytical solutions based on the strong-shock assumption. Qualitative or quantitative agreements were obtained for all these cases.</p>