  This paper is concerned with a non-conserved phase field system of Caginalp
type in which the main operators are fractional versions of two fixed linear
operators $A$ and $B$. The operators $A$ and $B$ are supposed to be densely
defined, unbounded, self-adjoint, monotone in the Hilbert space $L^2(\Omega)$,
for some bounded and smooth domain $\Omega$, and have compact resolvents. Our
definition of the fractional powers of operators uses the approach via spectral
theory. A nonlinearity of double-well type occurs in the phase equation and
either a regular or logarithmic potential, as well as a non-differentiable
potential involving an indicator function, is admitted in our approach. We show
general well-posedness and regularity results, extending the corresponding
results that are known for the non-fractional elliptic operators with zero
Neumann conditions or other boundary conditions like Dirichlet or Robin ones.
Then, we investigate the longtime behavior of the system, by fully
characterizing every element of the $\omega$-limit as a stationary solution. In
the final part of the paper we study the asymptotic behavior of the system as
the parameter $\sigma$ appearing in the operator $B^{2\sigma}$ that plays in
the phase equation decreasingly tends to zero. We can prove convergence to a
phase relaxation problem at the limit, in which an additional term containing
the projection of the phase variable on the kernel of $B$ appears.
