The QCD axion solving the strong CP problem may originate from antisymmetric tensor gauge fields in compactified string theory, with a decay constant around the GUT scale. Such possibility appears to be ruled out now by the detection of tensor modes by BICEP2 and the PLANCK constraints on isocurvature density perturbations. A more interesting and still viable possibility is that the string theoretic QCD axion is charged under an anomalous U(1)(A) gauge symmetry. In such case, the axion decay constant can be much lower than the GUT scale if moduli are stabilized near the point of vanishing Fayet-Illiopoulos term, and U(1) (A) -charged matter fields get a vacuum value v similar to (m(SUSY)M(Pl)(n) )(1/(n+1)) (n >= 0) induced by a tachyonic SUSY breaking mass m(SUSY). We examine the symmetry breaking pattern of such models during the inflationary epoch with H-I similar or equal to 10(14) GeV, and identify the range of the QCD axion decay constant, as well as the corresponding relic axion abundance, consistent with known cosmological constraints. In addition to the case that the PQ symmetry is restored during inflation, i.e. v(t(I)) = 0, there are other viable scenarios, including that the PQ symmetry is broken during inflation with v(t(I)) similar to (4 pi HIMPln)(1/(n+1)) similar to 10(16)-10(17) GeV due to the Hubble-induced D-term D-A similar to 8 pi H-2(I)2, while v(t(0)) similar to (m(SUSY) M-Pl(n))(1/(n+1)) similar to 10(9)-5 x 10(13) GeV in the present universe, where v(t(0)) above 10(12) GeV requires a fine-tuning of the axion misalignment angle. We also discuss the implications of our results for the size of SUSY breaking soft masses.