Nonlinear parabolic problems with lower order terms and related integral estimates

We deal with the solutions to nonlinear parabolic equations of the form $$ u_t-div a(x,t,Du)+g(x,t,u)=f(x,t) on \Omega_T=\Omega\times(-T,0), $$ under standard growth conditions on $g$ and $a$, with $f$ only assumed to be integrable to the power $\gamma > 1$. We prove general local decay estimates for level sets of the solutions $u$ and the gradient $Du$ which imply very general estimates in rearrangement function spaces (Lebesgue, Orlicz, Lorentz) and non-rearrangement ones, up to Lorentz–Morrey spaces.