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Peaking self-similar solutions and continuation after blow-up for a quasilinear Hardy-Hénon equation
Abstract
We consider the quasilinear Hardy-Hénon equation ut=Δum+|x|σup, (x,t)∈ℝN×(0,T), in the range of exponents and dimension N≥3, σ∈(−2,∞), m>1, p>pS(σ)=(m(N+2σ+2))/(N−2). We establish sharp ranges of exponents (m,p,σ) guaranteeing existence of \emph{peaking solutions} in backward self-similar form u(x,t)=(T−t)−αf(|x|(T−t)−β). By peaking solutions we understand solutions blowing up at a finite time T∈(0,∞) but allowing for a continuation after the blow-up time. As a remarkable outcome, a number of ranges of exponents σ and m for which peaking solutions exist for any p>pS(σ) are deduced, contrasting to the previously considered exponent σ=0, where a Lepin-type critical exponent pL provides an upper bound for p.
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BibTeX
@misc{iagar2026peaking,
title = {{Peaking self-similar solutions and continuation after blow-up for a quasilinear Hardy-Hénon equation}},
author = {Razvan Gabriel Iagar and Ariel Sánchez and Erik Sarrión-Pedralva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32304},
eprint = {2610.32304},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32304},
}
RIS
TY - UNPB
TI - Peaking self-similar solutions and continuation after blow-up for a quasilinear Hardy-Hénon equation
AU - Razvan Gabriel Iagar
AU - Ariel Sánchez
AU - Erik Sarrión-Pedralva
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32304
UR - https://arxiv.org/abs/2610.32304
AB - We consider the quasilinear Hardy-Hénon equation u_t=Δu^m+|x|^σu^p, (x,t)∈ℝ^N×(0,T), in the range of exponents and dimension N≥3, σ∈(−2,∞), m>1, p>p_S(σ)=(m(N+2σ+2))/(N−2). We establish sharp ranges of exponents (m,p,σ) guaranteeing existence of \emph{peaking solutions} in backward self-similar form u(x,t)=(T−t)^−αf(|x|(T−t)^−β). By peaking solutions we understand solutions blowing up at a finite time T∈(0,∞) but allowing for a continuation after the blow-up time. As a remarkable outcome, a number of ranges of exponents σ and m for which peaking solutions exist for any p>p_S(σ) are deduced, contrasting to the previously considered exponent σ=0, where a Lepin-type critical exponent p_L provides an upper bound for p.
ER -