Second critical exponent for a quasilinear Hardy--Hénon equation
Abstract
We study the Cauchy problem for the parabolic Hardy--Hénon equation ∂tu=Δum+|x|σup with slow diffusion m>1 and a potential which is singular at the origin, −2<σ<0, in dimension N≥3. Above the Fujita exponent, global and blowing-up solutions coexist, and we show that the spatial decay of the initial data decides between them. The second critical exponent turns out to be the decay rate of an explicit singular steady state. Data decaying more slowly blow up in finite time, whatever their size, while faster decaying data which are small in a weighted supremum norm give rise to global solutions. At the critical decay, and in the range where the singular steady state exists, we construct bounded stationary barriers with exactly critical tail by gluing the singular steady state to regular steady states. We deduce that the critical amplitude separating global existence from blow-up lies between the amplitude of the singular steady state and an explicit constant. We also show that data lying below a proper fraction of the singular steady state produce global bounded solutions, and that the singular amplitude itself is attained by global solutions whenever a Joseph--Lundgren type condition fails. Conjectures and open problems complete the picture.
Related
- Systems of Kirchhoff quasilinear Schrodinger equations with Hardy potential and critical Choquard-type nonlinearity
- Peaking self-similar solutions and continuation after blow-up for a quasilinear Hardy-Hénon equation
- Local Nevanlinna embedding for Hardy space of Dirichlet series below critical exponent
- The Hénon equation beyond the critical Sobolev exponent
- The critical exponent for the two-dimensional semilinear wave equation with strong damping
BibTeX
@misc{belgacem2026second,
title = {{Second critical exponent for a quasilinear Hardy--Hénon equation}},
author = {Rihab Ben Belgacem and Mohamed Majdoub},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06318},
eprint = {2611.06318},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06318},
}
RIS
TY - UNPB
TI - Second critical exponent for a quasilinear Hardy--Hénon equation
AU - Rihab Ben Belgacem
AU - Mohamed Majdoub
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06318
UR - https://arxiv.org/abs/2611.06318
AB - We study the Cauchy problem for the parabolic Hardy--Hénon equation ∂_tu=Δu^m+|x|^σu^p with slow diffusion m>1 and a potential which is singular at the origin, −2<σ<0, in dimension N≥3. Above the Fujita exponent, global and blowing-up solutions coexist, and we show that the spatial decay of the initial data decides between them. The second critical exponent turns out to be the decay rate of an explicit singular steady state. Data decaying more slowly blow up in finite time, whatever their size, while faster decaying data which are small in a weighted supremum norm give rise to global solutions. At the critical decay, and in the range where the singular steady state exists, we construct bounded stationary barriers with exactly critical tail by gluing the singular steady state to regular steady states. We deduce that the critical amplitude separating global existence from blow-up lies between the amplitude of the singular steady state and an explicit constant. We also show that data lying below a proper fraction of the singular steady state produce global bounded solutions, and that the singular amplitude itself is attained by global solutions whenever a Joseph--Lundgren type condition fails. Conjectures and open problems complete the picture.
ER -