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Backward estimates for nonnegative solutions to a class of singular parabolic equations

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 ([formula presented]<p<2) equations. Let us consider nonnegative solutions defined in R+×RN. We recall that in Recalde and Vespri (2015), starting from the value attained in a point (x0,t0) by the solution, we proved sharp estimates in the stripe ((1−ε)t0,∞)×RN. In this note we show that, quite surprisingly, sharp estimates hold also for the remote past i.e. also for the stripe (0,t0)×RN. 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. ([formula presented])+<m<1.