Sharp decay rate and asymptotic simplification for a quasilinear heat equation
Abstract
We study the decay rate for a quasilinear heat equation of the form $ \dot u-\Delta u = \operatorname{div B}(\nabla u)$. The term on the right-hand-side is interpreted as a nonlinear perturbation of the ordinary heat equation and we will prove an optimal decay rate for the solution. More precisely, under natural conditions on the operator B : Rd → Rd, we show that for sufficiently regular initial data the solution decays as t−d/4 and its gradient as t−d/4 − 1/2 for large times, that is at the same rates as for the free heat equation. Furthermore, we prove that the difference between the quasilinear and the linear solution decays \emph{strictly faster} than the solution of either equation generically does, that is that there is \emph{asymptotic simplification}. All estimates are completely explicit and rely on variations of Fourier splitting techniques, originally introduced by Schonbek.
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BibTeX
@misc{hajjar2026sharp,
title = {{Sharp decay rate and asymptotic simplification for a quasilinear heat equation}},
author = {Maryam Al Hajjar and Ioana Ciotir and Matthias Täufer},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37662},
eprint = {2609.37662},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05769211v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp decay rate and asymptotic simplification for a quasilinear heat equation
AU - Maryam Al Hajjar
AU - Ioana Ciotir
AU - Matthias Täufer
PY - 2026
DA - 2026/09/29
PB - HAL
DO - 10.48550/arxiv.2609.37662
UR - https://hal.science/hal-05769211v1
AB - We study the decay rate for a quasilinear heat equation of the form $ \dot u-\Delta u = \operatorname{div B}(\nabla u)$. The term on the right-hand-side is interpreted as a nonlinear perturbation of the ordinary heat equation and we will prove an optimal decay rate for the solution. More precisely, under natural conditions on the operator B : R^d → R^d, we show that for sufficiently regular initial data the solution decays as t^−d/4 and its gradient as t^−d/4 − 1/2 for large times, that is at the same rates as for the free heat equation. Furthermore, we prove that the difference between the quasilinear and the linear solution decays \emph{strictly faster} than the solution of either equation generically does, that is that there is \emph{asymptotic simplification}. All estimates are completely explicit and rely on variations of Fourier splitting techniques, originally introduced by Schonbek.
ER -