Global strong solvability of a quasilinear subdiffusion problem

TL;DR

This paper establishes the global strong solvability of a quasilinear subdiffusion problem with fractional time derivatives, relevant for modeling anomalous diffusion in materials with memory, using advanced regularity results.

Contribution

It proves the global strong solvability for a class of fractional quasilinear diffusion equations, extending the mathematical understanding of such models.

Findings

Proves global strong solvability of the problem.

Establishes interior Hölder continuity of solutions.

Provides an $L_2$ decay estimate for specific cases.

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.