This paper mainly deals with the multidimensional hydrodynamic model for semiconductors. Inspired by the previous papers [Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. we develop some new frequency-localization estimates to establish the global existence and exponential stability of (small) classical solutions in a class of critical Besov spaces, which are different from estimates in our recent paper [D.Y. Fang, J. Xu, T. Zhang, Global exponential stability of classical solutions to the hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci. 17 (2007Sci. 17 ( ) 1507Sci. 17 ( -1530. Furthermore, this new method can also be applied to the multidimensional Euler equations with damping. The analytic tool used is the Littlewood-Paley decomposition.