We derive the asymptotic first-passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $\alpha \left(x\right)$ varies with position. On a bounded interval with an absorbing and a reflecting boundary, we show that the survival probability decays as $\mathrm{\Psi }\left(t\right)\sim C{t}^{-{\alpha }_{*}}/{[ln(t/\tau )]}^{\nu }$, where ${\alpha }_{*}$ is the minimum value of the fractional exponent and $\nu$ is determined by the location and shape of the minimum. The exponent $\nu$ is fixed by the location and order of the minimum, taking the value $1/k$ at a $k\mathrm{th}$-order interior minimum, 1 at the reflecting boundary and 2 at the absorbing boundary. For a constant fractional exponent, $\nu =0$, so the logarithmic factor distinguishes a continuously varying order field from a homogeneous medium. We recover both exponents from simulated first-passage times by a direct linear fit, with no amplitude or geometric input. We validate the theory against exact Laplace-space solutions and Monte Carlo simulations for linear and nonlinear profiles of $\alpha \left(x\right)$.

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• Anomalous diffusion
• Continuous-time random walk
• First passage problems