Self-similar spherical shock solution with sustained energy injection

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We present the generalization of the Sedov-Taylor self-similar strong spherical shock solution for the case of a central energy source varying in time, $E=A t^k$, where $A$ and $k$ are constants. The known Sedov-Taylor solution corresponds to a particular adiabatic case of $k=0$ or \emph{instant shock} with an instant energy source of the shock, $E=A$. The self-similar hydrodynamic flow in the nonadiabatic $k\neq0$ case exists only under the appropriate local entropy (energy) input which must be supported by some radiative mechanism from the central engine. The specific case of $k=1$ corresponds to a permanent energy injection into the shock, or injection shock with a central source of constant luminosity, $L=A$, $E=A t$. The generalized self-similar shock solution may be applied to astrophysical objects in which the duration of central source activity is longer than the shock expansion time, e.g. the early phase of SN explosions, strong wind from stars and young pulsars, non-steady spherical outflow from black holes and collapsing dense stellar clusters with numerous neutron star collisions.