关于Bandle、Levine和Zhang的零质量问题的研究

中文摘要 AI 辅助

我们研究非齐次半线性热方程\\[ u_t-\Delta u=|u|^p+w(x) \qquad\text{在 }(0,\infty)\times\mathbb{R}^N \\] 的临界行为，其中\\(N\geq3\\)，\\(p>1\\)，且\\(w\in L^1(\mathbb{R}^N)\\)是非平凡的且总质量为零：\\[ \int_{\mathbb{R}^N}w(x)\\,dx=0. \\] 这个零质量情形由Bandle、Levine和Zhang在2000年作为开放问题提出，此后一直未得到解决。我们通过证明对于\\[ 1<p\leq\frac{N}{N-2} \\] 不存在全局弱解，为该问题提供了完整解答。结合已知的关于足够小数据的超临界存在性结果，这表明区分不存在性与存在性区域的临界指数仍然是\\[ p_c=\frac{N}{N-2}， \\] 与正质量情形相同。

英文摘要

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.

发表机构

• Department of Mathematics, College of Science, King Saud University（国王沙特大学理学院数学系）

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