Analysis of mass controlled reaction–diffusion systems with nonlinearities having critical growth rates

Abstract We analyze semilinear reaction–diffusion systems that are mass controlled, and have nonlinearities satisfy critical growth rates. The under consideration only assumed to natural assumptions, namely the preservation of non-negativity a control total mass. It is proved in dimension one if (slightly super-) cubic rates then system has unique global classical solutions. Moreover, case dissipation, solution bounded uniformly time sup-norm. One key idea proof Hölder continuity gradient solutions parabolic equation with possibly discontinuous diffusion coefficients low regular forcing terms. When possesses additionally an entropy inequality, existence boundedness shown for satisfying intermediate sum condition, which significant generalization main this combine modified Gagliardo-Nirenberg inequality newly developed $$L^p$$ L p -energy method Fitzgibbon et al. (SIAM J Math Anal 53(6):6771–6803, 2021) Morgan Tang (Commun Contemp Math, 2022). This also allows us deal higher dimensions, recently been touched context controlled systems.