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

We define a suitable class PDG of functions bearing unbalanced energy estimates, that are embodied by local weak subsolutions to doubly nonlinear, double-phase, Orlicz-type and fully anisotropic operators. Then we prove that members of PDG are locally bounded, under critical, sub-critical and limit growth conditions typical of singular and degenerate parabolic operators, with quantitative point-wise estimates that follow the lines of the pioneering work of Ladyzhenskaya, Solonnikov and Uraltseva [31] . These local bounds are new in the critical case and sub-critical cases, and have been obtained without any qualitative boundedness assumption. In particular, our proof of local boundedness in the critical case is valid disregarding of any additional integrability conditions and covers both the classical p -Laplacian and the porous medium equations.