PreprintCC BY 4.0arXiv
On the Zero-Mass Problem of Bandle, Levine, and Zhang
Abstract
We study the critical behavior of the inhomogeneous semilinear heat equation ut−Δu=|u|p+w(x) in (0,∞)×ℝN, where N≥3, p>1, and w∈ L1(ℝN) is nontrivial and has zero total mass: ∫ℝNw(x) dx=0. This zero-mass case was posed as an open problem by Bandle, Levine, and Zhang in 2000 and has remained unresolved since then. We provide a complete answer to this problem by proving nonexistence of global weak solutions for 1<p≤N/(N−2). Combined with the known supercritical existence result for sufficiently small data, this shows that the critical exponent separating the nonexistence and existence regimes remains pc=N/(N−2), the same as in the positive-mass case.
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BibTeX
@misc{jleli2026zeromass,
title = {{On the Zero-Mass Problem of Bandle, Levine, and Zhang}},
author = {Mohamed Jleli and Bessem Samet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22458},
eprint = {2609.22458},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22458},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Zero-Mass Problem of Bandle, Levine, and Zhang
AU - Mohamed Jleli
AU - Bessem Samet
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22458
UR - https://arxiv.org/abs/2609.22458
AB - We study the critical behavior of the inhomogeneous semilinear heat equation u_t−Δu=|u|^p+w(x) in (0,∞)×ℝ^N, where N≥3, p>1, and w∈ L^1(ℝ^N) is nontrivial and has zero total mass: ∫_ℝ^Nw(x) dx=0. This zero-mass case was posed as an open problem by Bandle, Levine, and Zhang in 2000 and has remained unresolved since then. We provide a complete answer to this problem by proving nonexistence of global weak solutions for 1<p≤N/(N−2). Combined with the known supercritical existence result for sufficiently small data, this shows that the critical exponent separating the nonexistence and existence regimes remains p_c=N/(N−2), the same as in the positive-mass case.
ER -