Abstract: In this paper we will study the wellposed of the initial value problem associated to the homogeneous left invariant differential operator on the Heisenberg group by the method of group Forier anasis ...  where a,>0,r∈R)on the Heisenberg group,then obtain the explicit expression of the fundamental solutions for the generalized heat operator  and the operator L.

Key words:  , group Fourier transform, fundamental solution, wellopsed