Global estimates for nonlinear parabolic equations

We consider nonlinear parabolic equations of the type $$ u_t - \text{div}\, a(x, t, Du)= f(x,t) \ \, \text{on} \ \Omega_T = \Omega\times (-T,0), $$ under standard growth conditions on $a$, with $f$ only assumed to be integrable. We prove general decay estimates up to the boundary for level sets of the solutions $u$ and the gradient $Du$ which imply very general estimates in Lebesgue and Lorentz spaces. Assuming only that the involved domains satisfy a mild exterior capacity density condition, we provide global regularity results.