Nonlinear parabolic problems with lower order terms and related integral estimates

We deal with the solutions to nonlinear parabolic equations of the form ut−diva(x,t,Du)+g(x,t,u)=f(x,t)onΩT=Ω×(−T,0), under standard growth conditions on gg and aa, with ff only assumed to be integrable to the power γ>1γ>1. We prove general local decay estimates for level sets of the solutions uu and the gradient DuDu which imply very general estimates in rearrangement function spaces (Lebesgue, Orlicz, Lorentz) and non-rearrangement ones, up to Lorentz–Morrey spaces.