Strong gradient absorption and initial decay prevent blow-up
Abstract
Consider positive classical solutions of the Cauchy problem for ut−Δu=up−μ|∇ u|q, with sufficiently smooth initial data u0, in the strong absorption regime, namely q≥ p>1 and μ>0. It has remained a long standing open problem whether finite time blow-up can occur if u0 decays at space infinity. We here solve this problem, showing that the answer in negative. To this end, we establish a novel, universal Bernstein-type gradient estimate, of independent interest, valid only in the range q≥ p. Moreover, the decay assumption on u0 turns out to be essentially optimal. Indeed we show that the solution blows up in finite time whenever u0 has a positive limit at infinity, and we determine the blow-up behavior of u at space infinity under an additional assumption.
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BibTeX
@misc{souplet2026strong,
title = {{Strong gradient absorption and initial decay prevent blow-up}},
author = {Philippe Souplet and Berikbol T. Torebek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28913},
eprint = {2610.28913},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28913},
}
RIS
TY - UNPB
TI - Strong gradient absorption and initial decay prevent blow-up
AU - Philippe Souplet
AU - Berikbol T. Torebek
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28913
UR - https://arxiv.org/abs/2610.28913
AB - Consider positive classical solutions of the Cauchy problem for u_t−Δu=u^p−μ|∇ u|^q, with sufficiently smooth initial data u_0, in the strong absorption regime, namely q≥ p>1 and μ>0. It has remained a long standing open problem whether finite time blow-up can occur if u_0 decays at space infinity. We here solve this problem, showing that the answer in negative. To this end, we establish a novel, universal Bernstein-type gradient estimate, of independent interest, valid only in the range q≥ p. Moreover, the decay assumption on u_0 turns out to be essentially optimal. Indeed we show that the solution blows up in finite time whenever u_0 has a positive limit at infinity, and we determine the blow-up behavior of u at space infinity under an additional assumption.
ER -