  We investigate whether the Affleck-Dine (AD) mechanism works when the
contribution of the two-loop thermal correction to the potential is negative in
the gauge-mediated supersymmetry breaking models. The AD field is trapped far
away from the origin by the negative thermal correction for a long time until
the temperature of the universe becomes low enough. The most striking feature
is that the Hubble parameter becomes much smaller than the mass scale of the
radial component of the AD field, during the trap. Then, the amplitude of the
AD field decreases so slowly that the baryon number is not fixed even after the
onset of radial oscillation. The resultant baryon asymmetry crucially depends
on whether the Hubble parameter, $H$, is larger than the mass scale of the
phase component of the AD field, $M_\theta$, at the beginning of oscillation.
If $H < M_\theta$ holds, the formation of Q balls plays an essential role to
determine the baryon number, which is found to be washed out due to the
nonlinear dynamics of Q-ball formation. On the other hand, if $H > M_\theta$
holds, it is found that the dynamics of Q-ball formation does not affect the
baryon asymmetry, and that it is possible to generate the right amount of the
baryon asymmetry.
