The dual-phase-lagging heat conduction equation is shown to be well-posed in a finite 1D region under Dirichlet, Neumann or Robin boundary conditions. Two solution structure theorems are developed for dual-phase-lagging heat conduction equations under linear boundary conditions. These theorems express contributions (to the temperature held) of the initial temperature distribution and the source term by that of the initial time-rate change of the temperature. This reveals the structure of the temperature field and considerably simplifies the development of solutions of dual-phase-lagging heat conduction equations.