A structural theorem for a vector valued exponentially bounded distribution is used for introducing and studyng of a class distribution semigroups. An infinitesimal generator of such a semigroup is not necessarily densely defined, but if it is the case, then it corresponds to a distribution semigroup introduced by Lions. This result is obtained by Wang and Kunstmann for a class of exponentially bounded quasi-distribution semigroups. In fact we show that our class of distribution semigroup is identical to Wang-Kunstmann's one. Our approach is completely different and gives new characterizations. Applications to equations $\displaystyle \frac{\partial u}{\partial t} = Au +f,$ where $A$ is not necessarily densely defined and $f$ is an exponential vector valued distribution supported by $[0, \infty ),$ are given. This paper is written much before the publishing of Wang's and Kunstmann's paper but because of various reasons it is published with a very long delay. Here it is given in the primary version as an original approach although some parts are consequences of published results of Wang and Kunstmann.