In this study, a number of notable physical anomalies concerning non-Fourier heat conduction under the dual-phase-lag (DPL) model arc observed and investigated. It is found that, during the transient heat transfer process, the over-diffusion mode predicts a "hyperactive" to "under-active" transition in thermal behavior. The main cause behind it lies in the time-varying effect of tau(T) (the phase lag of the temperature gradient) on the thermal response. Also, change of polarity in reflected thermal waves can be observed when a consta ant-temperature boundary is involved, which hints that a heating process may be followed by a spontaneous cooling effect. A fairly Strong connection is present between the tau(T)-induced dispersive effect and an unusual thermal accumulation phenomenon in an on-off periodic heating process. Furthermore, a paradox involving a moving medium is detected in the DPL model, which call be solved by replacing the temporal partial derivatives in the DPL equation with the material derivatives. During the process of analysis, a high-order characteristics-based TVD scheme is relied on to provide accurate and reliable numerical simulations to the DPL heat conduction equation under various initial-boundary conditions. (C) 2007 Elsevier Ltd. All rights reserved.