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

Encounter-based reaction-subdiffusion model II: partially absorbing traps and the occupation time propagator