Global regularity of solutions to systems of reaction-diffusion with Sub-Quadratic Growth in any dimension

TL;DR

This paper proves the global boundedness and regularity of solutions to reaction-diffusion systems with subquadratic growth in any dimension, using blow-up techniques and entropy methods, without smallness restrictions.

Contribution

It establishes the regularity of reaction-diffusion systems with subquadratic growth in all dimensions, extending previous results by removing smallness assumptions and employing novel entropy-based techniques.

Findings

Solutions are globally bounded and regular in any dimension.

Entropy methods enable control of blow-ups despite supercritical entropy.

The approach adapts De Giorgi methods to reaction-diffusion systems.

Abstract

This paper is devoted to the study of the regularity of solutions to some systems of reaction--diffusion equations, with reaction terms having a subquadratic growth. We show the global boundedness and regularity of solutions, without smallness assumptions, in any dimension $N$. The proof is based on blow-up techniques. The natural entropy of the system plays a crucial role in the analysis. It allows us to use of De Giorgi type methods introduced for elliptic regularity with rough coefficients. In spite these systems are entropy supercritical, it is possible to control the hypothetical blow-ups, in the critical scaling, via a very weak norm. Analogies with the Navier-Stokes equation are briefly discussed in the introduction.