Local well-posedness, blow-up phenomenon and ill-posedness for a model system of the radiating gas

In this paper, we investigate the Cauchy problem for a model system of radiating gas in multi-dimensional space. Firstly, we prove that the model is locally well-posed in Besov spaces $B_{p,r}^s({\mathbb{R}}^d)$ with $s>1+\frac{d}{p}$ (resp., $s=1+\frac{d}{p}$, $r=1$), including the endpoint space $B_{\mathrm{\infty },1}^1({\mathbb{R}}^d)$. Secondly, we show that the data-to-solution map is not uniformly continuous on bounded subsets of these Besov spaces. Moreover, we construct a blow-up scenario and derive two sufficient blow-up conditions, which improve the previous blow-up condition obtained in \cite{MR3016436}. Finally, we prove that the Cauchy problem is ill-posed in the critical space $B_{p,r}^{1+\frac{d}{p}}({\mathbb{R}}^d)$ for $1\le p\le \mathrm{\infty }$ and $r>1$ in the sense of norm inflation.