Time-fractional quenching problem: Blow-up of $D_t^{\alpha }u$ at the quenching point

In this paper, we establish the occurrence of blow-up in the time-fractional term at the quenching point. By demonstrating that the quenching points are contained within a compact subset of the designated spatial interval, we employ this finding to prove the blow-up of the Caputo fractional time-derivative at the quenching point.