Global strong solvability of a quasilinear subdiffusion problem

Abstract

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.