A Bernoulli-Type Localization Principle for Nonlinear Diffusion Equations: Structural Description of Solutions and First Application to Image Processing

Abstract

This paper develops a new method for analyzing nonlinear diffusion equations of porous--medium type with time-dependent growth rates, based on the \emph{localization of solutions} through an associated Bernoulli-type ordinary differential equation. This \emph{Bernoulli localization principle} separates the temporal and spatial components of the dynamics: the time evolution is governed explicitly by a Bernoulli ODE with a closed-form solution, while the spatial profile is determined by a stationary sublinear elliptic equation of Brezis--Oswald type. This decomposition replaces traditional qualitative analysis of the full parabolic PDE with a transparent combination of a scalar ODE and an elliptic problem, yielding a complete structural description of all solutions. We establish three main results: (i) existence and uniqueness of separable classical solutions obtained via the Bernoulli transformation; (ii) a comparison principle ensuring that orderings of initial data and growth rates persist in time; and (iii) existence and uniqueness of general, non-separable solutions through monotone iteration with explicit localization bounds derived from the Bernoulli ODE. To illustrate the applicability of the framework, we present what appears to be the first use of a porous-medium-type diffusion equation in digital image denoising, achieving performance comparable to or better than the Perona--Malik anisotropic diffusion filter, together with a complete Python implementation. The approach extends naturally to models in physics, biology, economics, and engineering, providing concrete predictions across a range of diffusion-driven phenomena.