In this paper, we consider a coupled parabolic system with multiple Henón-type components \(\left( \partial _t \textbf{u} - \Delta _{\mathbb {G}} \textbf{u} \right) (x,t) = H(x,t,\textbf{u})\) for \( (x,t) \in \mathbb {G} \times (0,T)\) , where \(\textbf{u}=(u_1,\cdots ,u_m) \) is the unknown, \(\mathbb {G}\) is a homogeneous Carnot group on \(\mathbb {R}^N\) , \(\Delta _{\mathbb {G}}\) is the operator whose components are given by the sub-Laplacian on \(\mathbb {G}\) , and \(\begin{aligned} H(x,t ,\textbf{u})=\left( t^{s_1} \ |x|_{\mathbb {G}}^{\gamma _1} \ u_2^{p_1}, t^{s_2} \ |x|_{\mathbb {G}}^{\gamma _2} \ u_3^{p_2}, \cdots , t^{s_{m}} \ |x|_{\mathbb {G}}^{\gamma _{m}} \ u_1^{p_{m}} \right) , \end{aligned}\) with \(p_i \ge 1\) , \(\textstyle \prod _{i=1}^m p_i>1\) , \(\gamma _i\ge 0\) , \(s_i> -1\) for \(i =1, \cdots , m\) , where \(|\cdot |_{\mathbb {G}}\) denotes a homogeneous norm on \(\mathbb {G}\) . We determine the Fujita-type exponent for this system, which depends on the homogeneous dimension Q of \(\mathbb {G}\) . In contrast to previous studies, our results are obtained by iterative methods involving the symmetric submarkovian semigroup associated with the operator \(\Delta _{\mathbb {G}}\) . In particular, the derivation of the blow-up result is delicate and is carried out through a novel approach based on Stirling’s asymptotic formula.
