Nonconcentration phenomenon for one-dimensional reaction–diffusion systems with mass dissipation

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 1, 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, that is, nonlinearities might have arbitrarily high growth rates, an additional entropy inequality had to be imposed. In this paper, 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 nonconcentration phenomenon for mass dissipating systems, that is the mass dissipation implies a dissipation in a Morrey space $M^{1,\delta }\left(\Omega \right)$ 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.