Global and local existence of solutions for nonlinear systems of time-fractional diffusion equations

TL;DR

This paper investigates the existence and blow-up of solutions for nonlinear two-component time-fractional diffusion systems with Neumann boundary conditions, using truncation and comparison principles.

Contribution

It introduces a method to handle all local Lipschitz nonlinearities with non-positive sum, establishing global existence and blow-up results.

Findings

Proves global existence of solutions under certain conditions.

Establishes conditions for solution blow-up.

Ensures non-negativity of solutions using comparison principles.

Abstract

In this paper, we consider initial-boundary value problems for two-component nonlinear systems of time-fractional diffusion equations with the homogeneous Neumann boundary condition and non-negative initial values. The main results are the existence of solutions global in time and the blow-up. Our approach involves the truncation of the nonlinear terms, which enables us to handle all local Lipschitz continuous nonlinear terms, provided their sum is less than or equal to zero. By employing a comparison principle for the corresponding linear system, we establish also the non-negativity of the nonlinear system.