  We propose a Hilbert space solution theory for a nonhomogeneous heat equation
with delay in the highest order derivatives with nonhomogeneous Dirichlet
boundary conditions in a bounded domain. Under rather weak regularity
assumptions on the data, we prove a well-posedness result and give an explicit
representation of solutions. Further, we prove an exponential decay rate for
the energy in the dissipative case. We also show that lower order
regularizations lead to ill-posedness, also for higher-order equations.
Finally, an application with physically relevant constants is given.
