  In this paper we develop an encounter-based model of reaction-subdiffusion in
a domain $\Omega$ with a partially absorbing interior trap $\calU\subset
\Omega$. We assume that the particle can freely enter and exit $\calU$, but is
only absorbed within $\calU$. We take the probability of absorption to depend
on the amount of time a particle spends within the trap, which is specified by
a Brownian functional known as the occupation time $A(t)$. The first passage
time (FPT) for absorption is identified with the point at which the occupation
time crosses a random threshold $\widehat{A}$ with probability density
$\psi(a)$. Non-Markovian models of absorption can then be incorporated by
taking $\psi(a)$ to be non-exponential. The marginal probability density for
particle position $\X(t)$ prior to absorption depends on $\psi$ and the joint
probability density for the pair $(\X(t),A(t))$, also known as the occupation
time propagator. In the case of normal diffusion one can use a Feynman-Kac
formula to derive an evolution equation for the propagator. However, care must
be taken when combining fractional diffusion with chemical reactions in the
same medium. Therefore, we derive the occupation time propagator equation from
first principles by taking the continuum limit of a heavy-tailed CTRW. We then
use the solution of the propagator equation to investigate conditions under
which the mean FPT (MFPT) for absorption within a trap is finite. We show that
this depends on the choice of threshold density $\psi(a)$ and the
subdiffusivity. Hence, as previously found for evanescent reaction-subdiffusion
models, the processes of subdiffusion and absorption are intermingled.
