  In this paper, we consider the boundary stabilization and observation of the
multidimensional unstable heat equation. Since we consider the heat equation in
a general domain, the usual partial differential equation back-stepping method
is hard to apply to the considered problems. The unstable dynamics of the heat
equation are treated by combining the finite-dimensional spectral truncation
method and the dynamics compensation method. By introducing additional
finite-dimensional actuator/sensor dynamics, the unbounded
stabilization/observation turns to be a bounded one. As a result, the
controller/observer design becomes more easier. Both the full state feedback
stabilizer and the state observer are designed. The exponential stability of
the closed-loop and the wellposedness of the observer are obtained.
