  We consider the multidimensional Euler-Poisson equations with non-zero heat
conduction, which consist of a coupled hyperbolic-parabolic-elliptic system of
balance laws. We make a deep analysis on the coupling effects and establish a
local well-posedness of classical solutions to the Cauchy problem pertaining to
data in the critical Besov space. Proof mainly relies on a standard iteration
argument. To achieve it, a new Moser-type inequality is developed by the Bony'
decomposition.

Analysis of PDEs