Reaction-diffusion systems with initial data of low regularity

TL;DR

This paper develops new methods to prove global existence of solutions for reaction-diffusion systems with low regularity initial data, simplifying proofs and extending results to super-quadratic cases and porous medium equations.

Contribution

It introduces an $L^1$ based a priori estimate for reaction-diffusion systems, enabling broader existence results and simplifying existing proofs.

Findings

Established $L^1$ initial data estimates for reaction-diffusion systems.

Proved new existence results for super-quadratic semilinear reactions.

Extended methods to semi-linear porous medium equations.

Abstract

Models issued from ecology, chemical reactions and several other application fields lead to semi-linear parabolic equations with super-linear growth. Even if, in general, blow-up can occur, these models share the property that mass control is essential. In many circumstances, it is known that this $L^1$ control is enough to prove the global existence of weak solutions. The theory is based on basic estimates initiated by M. Pierre and collaborators, who have introduced methods to prove $L^2$ a priori estimates for the solution. Here, we establish such a key estimate with initial data in $L^1$ while the usual theory uses $L^2$. This allows us to greatly simplify the proof of some results. We also establish new existence results of semilinearity which are super-quadratic as they occur in complex chemical reactions. Our method can be extended to semi-linear porous medium equations.