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arXiv · 2609.22458

On the Zero-Mass Problem of Bandle, Levine, and Zhang

Mohamed Jleli·Bessem Samet
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Abstract

We study the critical behavior of the inhomogeneous semilinear heat equation \[ u_t-Δu=|u|^p+w(x) \qquad\text{in }(0,\infty)\times\mathbb{R}^N, \] where \(N\geq3\), \(p>1\), and \(w\in L^1(\mathbb{R}^N)\) is nontrivial and has zero total mass: \[ \int_{\mathbb{R}^N}w(x)\,dx=0. \] 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 \[ 1<p\leq\frac{N}{N-2}. \] Combined with the known supercritical existence result for sufficiently small data, this shows that the critical exponent separating the nonexistence and existence regimes remains \[ p_c=\frac{N}{N-2}, \] the same as in the positive-mass case.

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Mohamed Jleli, Bessem Samet. 2026-09-18. On the Zero-Mass Problem of Bandle, Levine, and Zhang. https://arxiv.org/abs/2609.22458

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