Global estimates for nonlinear parabolic equations

TL;DR

This paper establishes comprehensive decay and regularity estimates for solutions to nonlinear parabolic equations with minimal assumptions on data and domain regularity, extending understanding of solution behavior in Lebesgue and Lorentz spaces.

Contribution

It introduces new decay estimates and global regularity results for nonlinear parabolic equations under weak conditions on data and domain geometry.

Findings

Decay estimates up to the boundary for solutions and gradients

Global regularity results under mild domain conditions

Estimates in Lebesgue and Lorentz spaces

Abstract

We consider nonlinear parabolic equations of the type $$ u_t - div a(x, t, Du)= f(x,t) on \Omega_T = \Omega\times (-T,0), $$ 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.