Reaction-Diffusion Group
Sections
ENRU
Menu
ENRU

Research group

Reaction-diffusion systems, uncertainty, and variational regularization

Reaction-Diffusion Group studies nonlinear reaction-diffusion and cross-diffusion systems, uncertainty-aware state models, and variational methods for biomedical imaging.

ResearchPublicationsJoin Us

Mission

The group studies mathematical structures that connect diffusion, reaction, uncertainty, and spatial regularization within a common analytical framework.

The aim is to keep proofs, models, and computation tightly coupled, so that theoretical control remains meaningful for biomedical and imaging applications.

Methodological profile

The methodology combines entropy methods, invariant-region arguments, regularity theory, and discrete-to-continuum analysis with variational formulations for segmentation and inverse problems.

Across projects, the focus is on how anisotropy, uncertainty, and discretization shape solvability, calibration, and continuum limits.

Scientific scope

Current work links rigorous nonlinear PDE analysis with uncertainty-aware modeling and variational schemes for tissue remodeling and medical image segmentation.

Key metrics

5

Featured 2026 papers

Papers and preprints currently highlighted on the site.

4

Research directions

From entropy-structured PDE analysis to uncertainty-aware imaging.

2

Application areas

Tissue modeling and medical image segmentation.

Research directions

Anisotropic reaction-diffusion models

PDE models with structured diffusion, uncertainty-aware state variables, and biomedical interpretation.

  • Myocardial remodeling
  • Tissue heterogeneity
  • Anisotropic biomedical transport

Cross-diffusion, entropy, and Markov kinetics

Entropy-structured analysis of coupled diffusion-reaction systems beyond the reach of classical maximum principles.

  • Multicomponent transport
  • Reaction networks
  • Entropy-consistent numerics

Dirichlet uncertainty models for imaging

Distributional representations of predictions and uncertainty for segmentation problems with spatial regularization.

  • Medical image segmentation
  • Uncertainty estimation
  • Calibration of pixelwise predictions

Scale-consistent variational regularization

Discrete-to-continuum analysis of edge-aware energies and structurally consistent anisotropic discretizations.

  • Edge-aware regularization
  • Inverse problems
  • Resolution-robust discretization

Featured projects

Active2026-present

Dirichlet reaction-diffusion model of myocardial remodeling

An anisotropic multicomponent tissue model that combines reaction-diffusion dynamics with uncertainty-aware Dirichlet concentration parameters.

Active2026-present

Entropy analysis of cross-diffusion systems with Markov kinetics

Entropy-based solvability, dissipation, and numerical analysis for coupled diffusion-reaction systems beyond the classical maximum-principle regime.

Active2025-present

Dirichlet-field segmentation with uncertainty estimation

Variational and neural segmentation methods that combine Dirichlet uncertainty fields with anisotropic spatial regularization.

Ongoing2026.0

Scale-consistent edge-aware regularization

Analysis of discrete weighted energies under mesh refinement and design of resolution-robust anisotropic regularizers.

Selected papers range from global solvability of cross-diffusion systems to anisotropic myocardial remodeling models and Dirichlet-field segmentation with scale-consistent regularization.

Selected results and publications

2026

On Global Solvability, Entropy Dissipation, and Numerical Analysis of Reaction-Diffusion Systems with Cross-Diffusion and Markov Kinetics

Zenodo preprint

Evgeny Yuryevich Shchetinin, Andrey Andreyevich Shevchuk

This paper studies multicomponent reaction-diffusion systems with cross-diffusion and Markov kinetics in an entropy-structured setting. It establishes global weak solvability for an admissible class, proves exponential relaxation in the conservative regime, and combines theory with entropy-consistent numerical experiments.

  • Nonlinear PDEs
  • Numerical Methods
PDF
2026

Global Existence, Invariant Region, and Enhanced Regularity for an Anisotropic Reaction-Diffusion Model of Myocardial Remodeling in Terms of Dirichlet Concentration Parameters

Zenodo preprint

Evgeny Yuryevich Shchetinin, Andrey Andreyevich Shevchuk

A multicomponent anisotropic reaction-diffusion model for post-infarction myocardial remodeling is formulated in terms of Dirichlet concentration parameters so that composition and uncertainty are represented together. The paper proves global existence, establishes invariant-region bounds, and derives enhanced regularity and short-time unconditional uniqueness under additional assumptions.

  • Nonlinear PDEs
  • Applied Systems
PDF
2026

Semantic Segmentation with Uncertainty Estimation Based on the Dirichlet Model and Anisotropic Regularization

Computational Mathematics and Information Technologies

Evgeny Yuryevich Shchetinin, Andrey Andreyevich Shevchuk

The paper proposes a semantic segmentation method that couples Dirichlet uncertainty modeling with anisotropic regularization. It proves Gamma-convergence and equicoercivity for the discrete energy family and reports calibrated uncertainty estimation with modest computational overhead.

  • Numerical Methods
  • Applied Systems
PDF
2026

On the Well-Posedness of Discrete Variational Problems with Edge-Aware Regularization

Modern Science: Theory and Practice

Evgeny Yuryevich Shchetinin, Andrey Andreyevich Shevchuk

This article analyzes why standard edge-aware weights collapse to isotropic behavior under grid refinement. It introduces a scale-consistent formulation based on normalized directional differences and proves convergence to a nontrivial weighted anisotropic Dirichlet energy, supported by numerical validation on synthetic and cardiac MRI data.

  • Nonlinear PDEs
  • Numerical Methods
PDF

People

Evgeny Yuryevich Shchetinin

Evgeny Yuryevich Shchetinin

Group Lead / Professor

Sevastopol State University

Evgeny works on nonlinear PDEs, cross-diffusion, entropy methods, and uncertainty-aware reaction-diffusion models with biomedical and imaging applications.

Reaction-diffusion systemsCross-diffusion and entropy methodsDirichlet uncertainty models
Andrey Andreyevich Shevchuk

Andrey Andreyevich Shevchuk

Researcher / PhD Student

Sevastopol State University

Andrey focuses on variational methods, anisotropic regularization, numerical analysis, and uncertainty estimation in medical image segmentation and reaction-diffusion modeling.

Variational regularizationNumerical analysis of PDEsMedical image segmentation

Materials

External ResourceNonlinear PDEsNumerical Methods

Cross-diffusion and Markov kinetics preprint

Open PDF version of the entropy-structured analysis of multicomponent cross-diffusion systems.

External ResourceNonlinear PDEsApplied Systems

Myocardial remodeling preprint

PDF copy of the anisotropic reaction-diffusion model for myocardial remodeling in Dirichlet concentration parameters.

External ResourceNumerical MethodsApplied Systems

Dirichlet-field segmentation paper

PDF resource covering Dirichlet uncertainty fields, uncertainty decomposition, and spatial regularizer discretization for segmentation.

News

NewsApr 20, 2026

Core 2026 papers added across four research directions

The site now brings together recent work on cross-diffusion, myocardial remodeling, Dirichlet uncertainty fields, and scale-consistent regularization.

Collaboration

The group welcomes collaboration on nonlinear PDEs, cross-diffusion, uncertainty quantification, variational imaging, and mathematically grounded numerical methods.

The site is intended as a concise entry point to the group's current papers, projects, and research questions.

View projectsContact the group