Abstract: In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient in the linear parabolic equation , with Dirichlet boundary conditions , . Main goal of this study is to investigate the distinguishability of the input–output mappings , via semigroup theory. In this paper, we show that if the null space of the semigroup consists of only zero function, then the input–output mappings and have the distinguishability property. Moreover, the values and of the unknown diffusion coefficient at and , respectively, can be determined explicitly by making use of measured output data (boundary observations) or/and . In addition to these, the values and of the unknown coefficient at and , respectively, are also determined via the input data. Furthermore, it is shown that measured output data and can be determined analytically, by an integral representation. Hence the input–output mappings , are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient at the end points and . [Copyright &y& Elsevier]