Mathematicians Prove Existence and Decay for Fractional Diffusion

Researchers Xinyue Zhang, Ning Pan, and Jiayu Sun have established the global existence and uniqueness of weak solutions for a specific class of fractional diffusion problems involving Kirchhoff-type nonlinearities. The work is presented as an early access article, providing immediate availability of peer-reviewed findings while the final version of record is prepared.

The mathematical problem under investigation is defined within a bounded domain \(\Omega\) in \(\mathbb{R}^{N}\) with a Lipschitz boundary. The core equation involves the Riemann-Liouville fractional time derivative \(\partial _{t}^{\beta}\), where the fractional order \(\beta\) satisfies \(0 < \beta < 1\), and the fractional Laplace operator \((-\Delta )^{\alpha}\), with \(\alpha\) constrained by \(0 < \alpha < 1\).

The boundary conditions specify that the solution vanishes outside the domain \(\Omega\), and the initial condition is given by a function \(u_{0}(x)\).

To prove the existence and uniqueness of weak solutions, the authors employed the Galerkin method. The validity of this existence result is contingent upon specific constraints on the nonlinearity exponent \(\rho\) relative to the spatial dimension \(N\) and the fractional order \(\alpha\). When the parameter \(\gamma\) is negative, the study requires \(0 < \rho < \frac{4\alpha}{N-2\alpha}\).

Beyond establishing existence, the research analyzes the long-term behavior of the solutions, specifically their decay properties. The authors utilized differential inequality techniques to derive estimates for how the solution diminishes over time.

Subject of Research: Existence and decay estimates for time-space fractional Kirchhoff-type diffusion problems

Article Title: Time-space fractional Kirchhoff-type diffusion problems: existence and decay estimates of solutions

Article References: Zhang, X., Pan, N., & Sun, J. (2026). Time-space fractional Kirchhoff-type diffusion problems: existence and decay estimates of solutions. Boundary Value Problems, 2026(1), Article 152. https://doi.org/10.1186/s13661-026-02350-5

Image Credits: AI Generated

DOI: 10.1186/s13661-026-02350-5

Keywords: fractional diffusion, Kirchhoff-type problems, partial differential equations, Galerkin method, decay estimates, weak solutions, fractional Laplace, Riemann-Liouville derivative, Boundary Value Problems, mathematical physics, Time-space, fractional