Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions

We introduce a suitable class PDG of functions characterized by unbalanced energy estimates, which arise from local weak subsolutions to doubly nonlinear, double phase, Orlicz type, and fully anisotropic operators. We prove that functions in PDG are locally bounded under critical, subcritical, and limiting growth conditions typical of singular and degenerate parabolic operators. Moreover, we establish quantitative pointwise estimates that follow the approach pioneered by Olga Ladyzhenskaya, Vsevolod Solonnikov, and Nina Uraltseva. These local boundedness results are new in both the critical and subcritical regimes and are obtained without any prior qualitative boundedness assumptions. In particular, our proof of local boundedness in the critical case does not require additional integrability conditions and applies, as special cases, to both the p-Laplacian equation and the porous medium equation.