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

We study the critical behavior of the inhomogeneous semilinear heat equation $$u_t-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^p+w(x)\text{in }(0,\mathrm{\infty })\times {\mathbb{R}}^N,$$ where $N\ge 3$, $p>1$, 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 $$1<p\le \frac{N}{N-2}\mathrm{.}$$ 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.