Work

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

Year: 2026

Type: article

Abstract: 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.

Source: Journal of Differential Equations

DOI: 10.1016/j.jde.2026.114235

Authors: Simone Ciani, Eurica Henriques, Mariia O. Savchenko, Igor I. Skrypnik

Institutions: University of Bologna, University of Trás-os-Montes and Alto Douro, Centro de Matemática Pólo da Universidade de Trás-os-Montes e Alto Douro, National Academy of Sciences of Ukraine

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550