Improved duality estimates and applications to reaction-diffusion equations

TL;DR

This paper introduces a refined duality estimate for parabolic equations, leading to new insights into reaction-diffusion systems, including smoothness, exponential convergence, and conditions for solutions in various dimensions.

Contribution

It provides a novel duality estimate and applies it to reaction-diffusion equations, establishing smoothness and convergence results not previously known.

Findings

Exponential convergence towards equilibrium in 2D systems.

Smooth solutions for reaction-diffusion systems from reversible chemistry.

Conditions for solutions when diffusion coefficients are close.

Abstract

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.