First-passage time for the g-subdiffusion process of vanishing particles

Subdiffusion equation and molecule survival equation, both with Caputo fractional time derivatives with respect to other functions g_{1} and g_{2}, respectively, are used to describe diffusion of a molecule that can disappear at any time with a constant probability. The process can be interpreted as an "ordinary" subdiffusion and "ordinary" molecule survival process in which timescales are changed by the functions g_{1} and g_{2}. We derive the first-passage time distribution for the process. The mutual influence of subdiffusion and molecule-vanishing processes can be included in the model when the functions g_{1} and g_{2} are related to each other. As an example, we consider the processes in which subdiffusion and molecule survival are highly related, which corresponds to the case of g_{1}≡g_{2}.