On the Well‐Posedness of a Nonlinear Diffusion Equation With Gray‐Level and Fractional Gradient Dependent Coefficients

ABSTRACT This paper presents a theoretical analysis of a nonlinear diffusion‐based model for image denoising. The proposed model introduces a diffusion coefficient that simultaneously depends on the image gradient, gray‐level information, and a fractional‐order gradient operator. The fractional derivative enhances the model's ability to preserve texture and structural features, while a gray‐level indicator contributes to effective noise suppression. A Gaussian kernel is integrated into the formulation to improve robustness against noise and ensure mathematical tractability. The model also includes a source term that enforces fidelity to the observed data. We establish the well‐posedness of the associated diffusion equation by proving the existences, uniqueness, and boundedness of the weak solution using Schauder's fixed point theorem.