Sharp lifespan estimates for heat equations with Hörmander operators
Abstract
In this paper, we study the Cauchy problem ut−ℒ u=h(t)up, u(0)=ε u0≥0, where p>1, h≥0 is only locally integrable, and ℒ=∑j=1mXj2 is a homogeneous Hörmander sum of squares on ℝn of homogeneous dimension q. Diffusion spreads the solution out and weakens the nonlinearity, while the reaction h(t)up may force blow-up. We prove that the outcome is decided by the single quantity ℋp,h(T)=∫0Th(t)(1+t)−q/2(p−1) d t, in which the reaction at time t is weighted by the factor by which diffusion has weakened the solution: small data are global if ℋp,h(∞)<∞, while otherwise the lifespan Tε satisfies c ε−(p−1)≤ℋp,h(Tε)≤ C ε−(p−1) for small ε. Our technique of proof is based on a backward-semigroup functional combined with a Gaussian-envelope energy and an ODE cap telescoped along interlaced geometric grids. The result contains the Euclidean, Heisenberg, Carnot and Grushin cases, yields explicit lifespan laws for power, logarithmic, exponential, periodic and intermittent coefficients, and exhibits effects with no autonomous counterpart, such as a lifespan with no power law in ε. We also prove regularity and a priori estimates for classical solutions and discuss what remains true for a general nonlinearity f(u).
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BibTeX
@misc{ruzhansky2026sharp,
title = {{Sharp lifespan estimates for heat equations with Hörmander operators}},
author = {Michael Ruzhansky and Berikbol T. Torebek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40547},
eprint = {2610.40547},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40547},
}
RIS
TY - UNPB
TI - Sharp lifespan estimates for heat equations with Hörmander operators
AU - Michael Ruzhansky
AU - Berikbol T. Torebek
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40547
UR - https://arxiv.org/abs/2610.40547
AB - In this paper, we study the Cauchy problem u_t−ℒ u=h(t)u^p, u(0)=ε u_0≥0, where p>1, h≥0 is only locally integrable, and ℒ=∑_j=1^mX_j^2 is a homogeneous Hörmander sum of squares on ℝ^n of homogeneous dimension q. Diffusion spreads the solution out and weakens the nonlinearity, while the reaction h(t)u^p may force blow-up. We prove that the outcome is decided by the single quantity ℋ_p,h(T)=∫_0^Th(t)(1+t)^−q/2(p−1) d t, in which the reaction at time t is weighted by the factor by which diffusion has weakened the solution: small data are global if ℋ_p,h(∞)<∞, while otherwise the lifespan T_ε satisfies c ε^−(p−1)≤ℋ_p,h(T_ε)≤ C ε^−(p−1) for small ε. Our technique of proof is based on a backward-semigroup functional combined with a Gaussian-envelope energy and an ODE cap telescoped along interlaced geometric grids. The result contains the Euclidean, Heisenberg, Carnot and Grushin cases, yields explicit lifespan laws for power, logarithmic, exponential, periodic and intermittent coefficients, and exhibits effects with no autonomous counterpart, such as a lifespan with no power law in ε. We also prove regularity and a priori estimates for classical solutions and discuss what remains true for a general nonlinearity f(u).
ER -