The piecewise Noetherian property in power series rings over a valuation domain

Let V be a valuation domain and let P be a nonzero prime ideal of V  . We characterize when V〚X〛P〚X〛V〚X〛P〚X〛 is a valuation domain and when it is a Noetherian ring. We then show that V〚X〛V〚X〛 is piecewise Noetherian if and only if V is Noetherian. As a corollary, we obtain that the piecewise Noetherian property is not preserved under the power series extension.