Let u be a non-negative super-solution to a 1-dimensional singular parabolic equation of p-Laplacian type ($1 < p < 2$). If u is bounded below on a time-segment $\{y\} \times (0; T] $by a positive number M, then it has a powerlike decay of order $\frac{p}{2-p}$ with respect to the space variable x in$ R \times[ T/2; T].$ This fact , is a "sidewise spreading of positivity" of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an eect is based on geometrical ideas.
1-DIMENSIONAL HARNACK ESTIMATES / Duzgun, Fatma Gamze; Gianazza, Ugo; Vespri, Vincenzo. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S. - ISSN 1937-1632. - STAMPA. - 9:(2016), pp. 675-685. [10.3934/dcdss.2016021]
1-DIMENSIONAL HARNACK ESTIMATES
VESPRI, VINCENZO
2016
Abstract
Let u be a non-negative super-solution to a 1-dimensional singular parabolic equation of p-Laplacian type ($1 < p < 2$). If u is bounded below on a time-segment $\{y\} \times (0; T] $by a positive number M, then it has a powerlike decay of order $\frac{p}{2-p}$ with respect to the space variable x in$ R \times[ T/2; T].$ This fact , is a "sidewise spreading of positivity" of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an eect is based on geometrical ideas.File | Dimensione | Formato | |
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