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

Subdiffusion equation and molecule survival equation, both with Caputo fractional time derivatives with respect to another 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 ``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 $g_1\equiv g_2$.