  We investigate the Cauchy problem for a heat equation involving a fractional
harmonic oscillator and an exponential nonlinearity. Our main contributions are
as follows: -We establish the local well-posedness in Orlicz spaces. -By
considering small initial data in suitable Lebesgue spaces, we derive the
existence of global weak-mild solutions. -We provide precise decay estimates
for large time, revealing that the decay rate depends on the behavior of the
nonlinearity near the origin. -Furthermore, we demonstrate that when
considering certain non-negative initial data within the appropriate Orlicz
space, the existence of local non-negative classical solutions is no longer
guaranteed. In summary, our work addresses the local and global behavior of
solutions, decay estimates, and the impact of nonlinearity on the existence of
classical solutions, offering some insights into the dynamics of the considered
heat equation with a fractional harmonic oscillator and exponential
nonlinearity.
