  Axion models have two serious cosmological problems, domain wall and
isocurvature perturbation problems. In order to solve these problems we
investigate the Linde's model in which the field value of the Peccei-Quinn (PQ)
scalar is large during inflation. In this model the fluctuations of the PQ
field grow after inflation through the parametric resonance and stable axionic
strings may be produced, which results in the domain wall problem. We study
formation of axionic strings using lattice simulations. It is found that in
chaotic inflation the axion model is free from both the domain wall and the
isocurvature perturbation problems if the initial misalignment angle \theta_a
is smaller than O(10^{-2}). Furthermore, axions can also account for the dark
matter for the breaking scale v \simeq 10^{12-16} GeV and the Hubble parameter
during inflation H_{inf} \lesssim 10^{11-12} GeV in general inflation models.
