Distribution semigroup in the sense of Wang and Kunstmana and the properties of infinitesimal generator are considered with in exponentially bounded distributions. Results are applied on a class of equations of the form $\displaystyle \frac{\partial }{\partial t} 4-An =f, \; f \in {\cal{K}}_{1}^{+} (L(e)),$ where $D(A) \subset L^{\infty }({\Bbb R})$ or $D(A) \subset E=C_{b}({\Bbb R}).$