  The standard problem for the classical heat equation posed in a bounded
domain $\Omega$ of $\mathbb R^n$ is the initial and boundary value problem. If
the Laplace operator is replaced by a version of the fractional Laplacian, the
initial and boundary value problem can still be solved on the condition that
the non-zero boundary data must be singular, i.e., the solution $u(t,x)$ blows
up as $x$ approaches $\partial \Omega$ in a definite way. In this paper we
construct a theory of existence and uniqueness of solutions of the parabolic
problem with singular data taken in a very precise sense, and also admitting
initial data and a forcing term. When the boundary data are zero we recover the
standard fractional heat semigroup. A general class of integro-differential
operators may replace the classical fractional Laplacian operators, thus
enlarging the scope of the work. As further results on the spectral theory of
the fractional heat semigroup, we show that a one-sided Weyl-type law holds in
the general class, which was previously known for the restricted and spectral
fractional Laplacians, but is new for the censored (or regional) fractional
Laplacian. This yields bounds on the fractional heat kernel.
