  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.
