COMPARISON PRINCIPLE FOR A NONLINEAR DIFFUSION EQUATION UNDER NONLOCAL BOUNDARY CONDITION

Abstract

In this paper, we study a Cauchy problem with a nonlocal Neumann boundary condition for a nonlinear time-space fractional diffusion equation involving the Caputo fractional derivative and the fractional p -Laplacian. The principal contribution of the paper is the establishment of a comparison principle for weak subsolutions and supersolutions under suitable assumptions on the parameters β, γ, k, and ℓ. This result shows that the ordering of the initial data and the corresponding nonlinear terms is preserved throughout the evolution, despite the presence of both temporal and spatial nonlocality. The proof requires a careful treatment of the Caputo derivative, the monotonicity properties of the fractional p-Laplacian, and the nonlocal Neumann boundary condition. The comparison principle then serves as the main analytical tool for investigating the qualitative behaviour of solutions. As direct applications, we derive a boundedness result in the dissipative case and establish a sufficient criterion for finite-time blow-up by comparing a weak solution with a suitable subsolution of an associated fractional differential equation.