Pointwise-in-time error bounds for semilinear and quasilinear fractional subdiffusion equations on graded meshes

Time-fractional semilinear and quasilinear parabolic equations with a Caputo time derivative of order $\alpha \in (0,1)$ are considered, solutions of which exhibit a singular behaviour at an initial time of type $t^{\sigma }$ for any fixed $\sigma \in (0,1)\cup (1,2)$. The L1 scheme in time is combined with a general class of discretizations for the semilinear term. For such discretizations, we obtain sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading. Both semi-discretizations in time and full discretizations using finite differences and finite elements in space are addressed. The theoretcal findings are illustrated by numerical experiments.