  We prove the global strong solvability of a quasilinear initial-boundary
value problem with fractional time derivative of order less than one. Such
problems arise in mathematical physics in the context of anomalous diffusion
and the modelling of dynamic processes in materials with memory. The proof
relies heavily on a regularity result about the interior H\"older continuity of
weak solutions to time fractional diffusion equations, which has been proved
recently by the author. We further establish a basic $L_2$ decay estimate for
the special case with vanishing external source term and homogeneous Dirichlet
boundary condition.

Analysis of PDEs