  This paper studies the well-posedness of a class of nonlocal parabolic
partial differential equations (PDEs), or equivalently equilibrium
Hamilton-Jacobi-Bellman equations, which has a strong tie with the
characterization of the equilibrium strategies and the associated value
functions for time-inconsistent stochastic control problems. Specifically, we
consider nonlocality in both time and space, which allows for modelling of the
stochastic control problems with initial-time-and-state dependent objective
functionals. We leverage the method of continuity to show the global
well-posedness within our proposed Banach space with our established Schauder
prior estimate for the linearized nonlocal PDE. Then, we adopt a linearization
method and Banach's fixed point arguments to show the local well-posedness of
the nonlocal fully nonlinear case, while the global well-posedness is
attainable provided that a very sharp a-priori estimate is available. On top of
the well-posedness results, we also provide a probabilistic representation of
the solutions to the nonlocal fully nonlinear PDEs and an estimate on the
difference between the value functions of sophisticated and na\"{i}ve
controllers. Finally, we give a financial example of time inconsistency that is
proven to be globally solvable.
