Refined blow-up behavior for semilinear heat equations with arbitrarily fast growing nonlinearities
Abstract
We study radially nonincreasing, positive blow-up solutions of a class of superlinear heat equations in the non-scale-invariant setting. This class contains nonlinearities with arbitrarily fast, super-exponential growth. Under suitable structural assumptions on the nonlinearity, we derive the sharp global blow-up profile in the original space-time variables. More precisely, for positive nonlinearities f satisfying limX→ ∞f'(X)G(X)= q≥1 where G(X)=∫X∞( ds)/f(s)<∞ ∀ X>0, and assuming either the time monotonicity ut≥0 holds or q>pS/(pS−1), we have u(x,t)=(1+o(1)) G−1(T−t+|x|2/(4q|log |x|2|)), as $(x,t)→ (0,T)$. This estimate in particular provides the sharp final space profile and the corresponding refined space-time profile. Moreover, our result recovers the universal structure of the global blow-up profile, given by the resolvent G−1 of the associated ODE evaluated at a time-space building block, which only depends on the limit q.
Related
- Confinement effects for semilinear fractional Hermite heat equations with nonlinearities of arbitrarily rapid growth
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- Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions
BibTeX
@misc{chabi2026refined,
title = {{Refined blow-up behavior for semilinear heat equations with arbitrarily fast growing nonlinearities}},
author = {Loth Damagui Chabi},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20013},
eprint = {2610.20013},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05801881v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Refined blow-up behavior for semilinear heat equations with arbitrarily fast growing nonlinearities
AU - Loth Damagui Chabi
PY - 2026
DA - 2026/10/14
PB - HAL
DO - 10.48550/arxiv.2610.20013
UR - https://hal.science/hal-05801881v1
AB - We study radially nonincreasing, positive blow-up solutions of a class of superlinear heat equations in the non-scale-invariant setting. This class contains nonlinearities with arbitrarily fast, super-exponential growth. Under suitable structural assumptions on the nonlinearity, we derive the sharp global blow-up profile in the original space-time variables. More precisely, for positive nonlinearities f satisfying lim_X→ ∞f'(X)G(X)= q≥1 where G(X)=∫_X^∞( ds)/f(s)<∞ ∀ X>0, and assuming either the time monotonicity u_t≥0 holds or q>p_S/(p_S−1), we have u(x,t)=(1+o(1)) G^−1(T−t+|x|^2/(4q|log |x|^2|)), as $(x,t)→ (0,T)$. This estimate in particular provides the sharp final space profile and the corresponding refined space-time profile. Moreover, our result recovers the universal structure of the global blow-up profile, given by the resolvent G^−1 of the associated ODE evaluated at a time-space building block, which only depends on the limit q.
ER -