Global solutions u to the semilinear heat equation \(\partial _tu-\Delta u=\mu |u|^{\lambda -l}u^l\) on \((0,\infty ) \times \mathbb {R}^n\) are considered, where \(n \ge 2\) , \(1+2/n<\lambda <\infty \) , \(\lambda \notin \mathbb {N}\) , \(l \in \{0,1\}\) , and \(\mu \in \mathbb {R} \setminus \{0\}\) . In the author’s preceding paper (Commun. Pure Appl. Anal. 23 (2024), no. 6, 830–872), it was shown that while \(u \in C((0,\infty );C^{\lambda +2-\sigma }(\mathbb {R}^n))\) holds for all \(0<\sigma <1\) , there exists an initial datum \(a=u(0,\cdot ) \in C_0^{\infty }(\mathbb {R}^n)\) such that \(u \notin C((0,\infty );C^{\lambda +2+\sigma }(\mathbb {R}^n))\) for any \(0<\sigma <1\) . In this paper, we consider the marginal case \(\sigma =0\) and show that while \(u \in C_{\textrm{w}}((0,\infty );C^{\lambda +2}(\mathbb {R}^n))\) holds, there exists an initial datum \(a=u(0,\cdot ) \in C_0^{\infty }(\mathbb {R}^n)\) such that \(u \notin L_{\textrm{loc}}^{\infty }((0,\infty );h_{\textrm{loc}}^{\lambda +2}(\mathbb {R}^n))\) and \(u \notin C((0,\infty );C_{\textrm{loc}}^{\lambda +2}(\mathbb {R}^n))\) , where \(h_{\textrm{loc}}^{\lambda +2}(\mathbb {R}^n)\) denotes the local little-Hölder spaces. Our results imply that in general, the regularity \(u \in C_{\textrm{w}}((0,\infty );C^{\lambda +2}(\mathbb {R}^n))\) is optimal in space and time.
