The linear version of the general theory of heat conduction is employed to study the problem of a half space subjected to step time inputs of temperature. The solution is obtained by the use of the Laplace transform on time and the sine transform on space. Exact solutions, wave-front, and long-time approximations are obtained. The long-time solutions satisfy the classical heat equation. It is shown that stable solutions may be obtained under more relaxed conditions than those prescribed by the general theory.