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Existence and Uniqueness for a Porous-Medium-Type Diffusion Model with Time-Dependent Growth

Dragos-Patru Covei

TL;DR

The paper studies a porous-medium-type nonlinear diffusion equation $\partial_t v - \Delta v^{\alpha} = \mu(t) v$ on a bounded domain with Dirichlet boundary conditions. It derives an explicit separable solution form $v_{\gamma}(x,t)=S(t)u^{1/\alpha}(x)$ where $u$ solves $-\Delta u = u^{1/\alpha}$ and $S$ obeys $S'(t)+S^{\alpha}(t)=\mu(t)S(t)$, decoupling space and time. Three main contributions are proved: (i) existence/uniqueness of separable solutions, (ii) a generalized comparison principle with respect to initial data and growth rate, and (iii) existence/uniqueness for non-separable solutions via a monotone-iteration subsolution–supersolution scheme. A digital image denoising application demonstrates robustness relative to the classical Perona–Malik model, highlighting practical impact and suggesting extensions to more complex geometries and boundary conditions.

Abstract

This paper investigates a nonlinear diffusion equation, characterized by a power-law dispersal mechanism and a time-dependent growth rate in a bounded domain with homogeneous Dirichlet boundary conditions. We establish three main theoretical contributions: first, the existence and uniqueness of separable classical solutions via a transformation into a stationary sublinear elliptic framework; second, a generalized comparison principle with respect to both initial conditions and growth rates; and third, a general existence and uniqueness result for non-separable solutions via a monotone iteration method. The connection between the temporal dynamics and the spatial profile is made explicit, providing analytical expressions for the solution. Furthermore, we demonstrate the model's effectiveness as a successful application in digital image denoising, showing superior robustness compared to the classical Perona--Malik model.

Existence and Uniqueness for a Porous-Medium-Type Diffusion Model with Time-Dependent Growth

TL;DR

The paper studies a porous-medium-type nonlinear diffusion equation on a bounded domain with Dirichlet boundary conditions. It derives an explicit separable solution form where solves and obeys , decoupling space and time. Three main contributions are proved: (i) existence/uniqueness of separable solutions, (ii) a generalized comparison principle with respect to initial data and growth rate, and (iii) existence/uniqueness for non-separable solutions via a monotone-iteration subsolution–supersolution scheme. A digital image denoising application demonstrates robustness relative to the classical Perona–Malik model, highlighting practical impact and suggesting extensions to more complex geometries and boundary conditions.

Abstract

This paper investigates a nonlinear diffusion equation, characterized by a power-law dispersal mechanism and a time-dependent growth rate in a bounded domain with homogeneous Dirichlet boundary conditions. We establish three main theoretical contributions: first, the existence and uniqueness of separable classical solutions via a transformation into a stationary sublinear elliptic framework; second, a generalized comparison principle with respect to both initial conditions and growth rates; and third, a general existence and uniqueness result for non-separable solutions via a monotone iteration method. The connection between the temporal dynamics and the spatial profile is made explicit, providing analytical expressions for the solution. Furthermore, we demonstrate the model's effectiveness as a successful application in digital image denoising, showing superior robustness compared to the classical Perona--Malik model.
Paper Structure (8 sections, 3 theorems, 82 equations, 1 table)

This paper contains 8 sections, 3 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of the Main Results
  3. Derivation of the Model via Conservation Laws
  4. Conclusions
  5. Python Implementation of Theorem \ref{['th1']}
  6. Python Implementation of Theorem \ref{['th2']}
  7. Python Implementation of Remark \ref{['rem']}
  8. Image Denoising Application and Comparison with Perona--Malik

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^{N}$ be a bounded, arcwise-connected open set with a compact closure and boundary $\partial \Omega$ of class $C^{3}$. Under these conditions, the nonlinear diffusion problem nd--bnd1 admits a unique solution $v_{\gamma}(x,t)$, given by 2, where $u\in C^{2}(\Omega) \cap

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • proof : Proof of Theorem \ref{['th1']}
  • proof : Proof of Theorem \ref{['th-comp']}
  • proof : Proof of Theorem \ref{['th2']}