We study the large time behavior of non-negative solutions to thenonlinear diffusion equation with critical gradient absorption$$\partial_t u-\Delta_{p}u+|\nabla u|^{q_*}=0 \quad \hbox{in} \(0,\infty)\times\mathbb{R}^N\ ,$$for $p\in(2,\infty)$ and $q_*:=p-N/(N+1)$. We show that theasymptotic profile of compactly supported solutions is given by asource-type self-similar solution of the $p$-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results.
Publié le : 2017-07-04
Classification:
large time behavior,
degenerate diffusion,
decay estimates,
gradient absorption,
logarithmic scales,
35K59, 35K65, 35K92, 35B40,
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
@article{hal-01135830,
author = {Gabriel Iagar, Razvan and Lauren\c cot, Philippe},
title = {Large time behavior for a quasilinear diffusion equation with critical gradient absorption},
journal = {HAL},
volume = {2017},
number = {0},
year = {2017},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-01135830}
}
Gabriel Iagar, Razvan; Laurençot, Philippe. Large time behavior for a quasilinear diffusion equation with critical gradient absorption. HAL, Tome 2017 (2017) no. 0, . http://gdmltest.u-ga.fr/item/hal-01135830/