Non-concentration phenomenon for one dimensional reaction-diffusion systems with mass dissipation

TL;DR

This paper proves a new non-concentration phenomenon in one-dimensional reaction-diffusion systems with mass dissipation, leading to global bounded solutions without requiring additional entropy conditions, and applies it to complex oscillatory systems.

Contribution

It introduces a novel non-concentration result for mass dissipating systems, removing the need for entropy assumptions in establishing global boundedness.

Findings

Mass dissipation implies Morrey space dissipation.

Global bounded solutions are achieved without extra entropy conditions.

Application to oscillatory Belousov-Zhabotinsky system.

Abstract

Reaction-diffusion systems with mass dissipation are known to possess blow-up solutions in high dimensions when the nonlinearities have super quadratic growth rates. In dimension one, it has been shown recently that one can have global existence of bounded solutions if nonlinearities are at most cubic. For the cubic intermediate sum condition, i.e. nonlinearities might have arbitrarily high growth rates, an additional entropy inequality had to be imposed. In this article, we remove this extra entropy assumption completely and obtain global boundedness for reaction-diffusion systems with cubic intermediate sum condition. The novel idea is to show a non-concentration phenomenon for mass dissipating systems, that is the mass dissipation implies a dissipation in a Morrey space $\mathsf{M}^{1,\delta}(\Omega)$ for some $\delta>0$. As far as we are concerned, it is the first time such a bound is derived for mass dissipating reaction-diffusion systems. The results are then applied to obtain global existence and boundedness of solutions to an oscillatory Belousov-Zhabotinsky system, which satisfies cubic intermediate sum condition but does not fulfill the entropy assumption. Extensions include global existence mass controlled systems with slightly-super cubic intermediate sum condition.