Well-Posedness of some inhomogeneous heat equation

We show the existence and uniqueness of the solution of inhomogeneous heat equations in Sobolev spaces if the external source also lives in similar spaces.
Energy of the solution is defined and used to show an explicit upper bound (called energy inequality) on Sobolev norms of the solution and its derivatives.
The analysis starts from Schwartz space functions and is generalized to Sobolev spaces by density argument