Optimal global well-posedness for semilinear heat equations
Abstract
We address the question of global in time existence for the semilinear parabolic equation { rcl ∂t u−Δu&=&|u|b−1u ~in ℝn× (0,∞), u(·,0)&=& u0 ~in ℝn, . where b> 1+2/n, n≥ 1. We found a space of initial data Z that guarantees the existence of a global in time solution whenever the norm of the initial datum u0 in Z is small. The space Z is optimal in the sense that if u0 is a nonnegative distribution and the equation admits a nonnegative solution u∈ Lbloc(ℝn×(0,∞)), then u0∈ Z. Moreover, our result contains all of the known results in the literature. In particular, it implies that global in time existence does hold for end-point Besov and Triebel-Lizorkin spaces, which resolves an open problem posed by L. Brandolese and F. Cortez (J. Funct. Anal. 276 (2019), 2589-2604).
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BibTeX
@misc{diebou2026optimal,
title = {{Optimal global well-posedness for semilinear heat equations}},
author = {Gael Y. Diebou and Nguyen Cong Phuc},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27365},
eprint = {2610.27365},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27365},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal global well-posedness for semilinear heat equations
AU - Gael Y. Diebou
AU - Nguyen Cong Phuc
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27365
UR - https://arxiv.org/abs/2610.27365
AB - We address the question of global in time existence for the semilinear parabolic equation { rcl ∂_t u−Δu&=&|u|^b−1u ~in ℝ^n× (0,∞), u(·,0)&=& u_0 ~in ℝ^n, . where b> 1+2/n, n≥ 1. We found a space of initial data Z that guarantees the existence of a global in time solution whenever the norm of the initial datum u_0 in Z is small. The space Z is optimal in the sense that if u_0 is a nonnegative distribution and the equation admits a nonnegative solution u∈ L^b_loc(ℝ^n×(0,∞)), then u_0∈ Z. Moreover, our result contains all of the known results in the literature. In particular, it implies that global in time existence does hold for end-point Besov and Triebel-Lizorkin spaces, which resolves an open problem posed by L. Brandolese and F. Cortez (J. Funct. Anal. 276 (2019), 2589-2604).
ER -