  The purpose in this paper is to determine the global behavior of solutions to
the initial-boundary value problems for energy-subcritical and critical
semilinear heat equations by initial data with lower energy than the mountain
pass level in energy spaces associated with self-adjoint operators satisfying
Gaussian upper bounds. Our self-adjoint operators include the Dirichlet
Laplacian on an open set, Robin Laplacian on an exterior domain, and
Schr\"odinger operators, etc.
