Preprint · arXiv · June 24, 2025

A Nonlinear Nonlocal Problem for the Caputo Fractional Subdiffusion Equation

Ravshan Ashurov, Rajapboy Saparboyev, Navbahor Nuraliyeva

Read on arXiv PDFDOI 10.48550/arxiv.2506.19516

Abstract

In this paper, we study a time-fractional subdiffusion equation with a nonlinear nonlocal initial condition involving the unknown solution at the final time. The considered problem is formulated using the Caputo fractional derivative of order \(0 < α< 1\), along with homogeneous Dirichlet boundary conditions. The nonlocal initial condition is of the form \( u(x,0) = g(x, u(x,T)) \), where \(g\) is a nonlinear function satisfying a Lipschitz condition. The main challenge arises from the implicit dependence on the unknown final state. Using an explicit representation of the solution in terms of the Green function and applying the Banach fixed point theorem, we establish the existence and uniqueness of a regular solution. We also provide uniform estimates for the Green function and analyze the influence of the Lipschitz constant on solvability.

Nonlinear Differential Equations AnalysisFractional Differential Equations SolutionsDifferential Equations and Boundary ProblemsCaputo fractional derivativefinal value problemexistence and uniqueness of solutions

A Nonlinear Nonlocal Problem for the Caputo Fractional Subdiffusion Equation

Abstract

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