  This paper is concerned with the Cauchy problem of the Burgers equation with
the critical dissipation. The well-posedness and analyticity in both of the
space and the time variables are studied based on the frequency decomposition
method. The large time behavior is revealed for any large initial data. As a
result, it is shown that any smooth and integrable solution is analytic in
space and time as long as time is positive and behaves like the Poisson kernel
as time tends to infinity. The corresponding results are also obtined for the
quasi-geostrophic equation.
