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

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