  We analyze semilinear reaction-diffusion systems that are mass controlled,
and have nonlinearities that satisfy critical growth rates. The systems under
consideration are only assumed to satisfy natural assumptions, namely the
preservation of non-negativity and a control of the total mass. It is proved in
dimension one that if nonlinearities have (slightly super-) cubic growth rates
then the system has a unique global classical solutions. Moreover, in the case
of mass dissipation, the solution is bounded uniformly in time in sup-norm. One
key idea in the proof is the H\"older continuity of gradient of solutions to
parabolic equation with possibly discontinuous diffusion coefficients and low
regular forcing terms. When the system possesses additionally an entropy
inequality, the global existence and boundedness of a unique classical solution
is shown for nonlinearities satisfying a cubic intermediate sum condition,
which is a significant generalization of cubic growth rates. The main idea in
this case is to combine a modified Gagliardo-Nirenberg inequality and the newly
developed $L^p$-energy method in
\cite{morgan2021global,fitzgibbon2021reaction}. This idea also allows us to
deal with the case of discontinuous diffusion coefficients in higher
dimensions, which has \blue{only recently been touched} in the context of mass
controlled reaction-diffusion systems.
