Backward estimates for nonnegative solutions to a class of singular parabolic equations. (October 2016) - Recalde, Marco Vinicio Calahorrano; Vespri, Vincenzo
Abstract: In this paper we improve a recent result of the same Authors (Recalde and Vespri, 2015) by using the same approach in a more tricky way. More precisely, we study a class of quasilinear singular parabolic equations with L ∞ coefficients, whose prototypes are the p -Laplacian ( 2 N N + 1 < p < 2 ) equations. Let us consider nonnegative solutions defined in R + × R N . We recall that in Recalde and Vespri (2015), starting from the value attained in a point ( x 0, t 0 ) by the solution, we proved sharp estimates in the stripe ( ( 1 − ε ) t 0, ∞ ) × R N . In this note we show that, quite surprisingly, sharp estimates hold also for the remote past i.e. also for the stripe ( 0, t 0 ) × R N . In the last section we briefly show how these results can be adapted to equations of Porous Medium type in the fast diffusion range i.e. ( N − 2 N ) + < m < 1 .