Calculus of Variations and Geometric Measure Theory

A. Di Castro - G. Palatucci

Nonlinear parabolic problems with lower order terms and related integral estimates

created by palatucci on 15 Mar 2012
modified on 19 Sep 2012


Published Paper

Inserted: 15 mar 2012
Last Updated: 19 sep 2012

Journal: Nonlinear Anal.
Volume: 75
Pages: 4177-4197
Year: 2012


We deal with the solutions to nonlinear parabolic equations of the form $$ ut - \text{div}\, a(x, t, Du) + g(x,t,u) = f(x,t) \ \, \text{on} \ \OmegaT = \Omega\times (-T,0), $$ 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.

Keywords: parabolic equations, Morrey-Lorentz regularity, Rearrangement function spaces, lower-order term, absorption term