On a nonlinear evolution equation with spatially variable growth arising in multi-frame image super-resolution modeling

Abstract

We study a nonlinear evolution equation with spatially variable growth arising from a regularized diffusion model for image restoration. The proposed model features a variable-exponent diffusion operator, allowing adaptive control of the diffusion intensity across the image domain, which enables effective noise suppression while preserving edges and fine structures. From a theoretical standpoint, we establish the existence of weak solutions by reformulating the problem as a fixed-point problem and applying Schauder's theorem, complemented by new a priori estimates tailored to the variable-exponent setting. We also provide numerical simulations to illustrate the qualitative behavior of solutions and to validate the model in image denoising and super-resolution tasks. The results demonstrate the ability of the proposed framework to achieve high-quality restorations, highlighting the interplay between rigorous analysis and practical imaging applications.