We are concerned with the existence of global and blow-up solutions for the nonlinear parabolic problem described by the Hardy-Hénon equation \(u_t - \Delta _{\mathbb {H}} u = |\cdot |_{\mathbb {H}}^{\gamma } u^p \text{ in } \mathbb {H}^N \times (0,T),\) where \(\mathbb {H}^N\) is the N-dimensional Heisenberg group, and the singular term \(|\cdot |_{\mathbb {H}}^{\gamma }\) is given by the Korányi norm. Our study focuses on nonnegative solutions. We establish that for \(\gamma \ge 0\) , the Fujita critical exponent is \(p_c = 1+ (2+\gamma )/Q\) , where \(Q=2N+2\) is the homogeneous dimension of \(\mathbb {H}^N\) . For \(-2<\gamma <0\) , the solutions blow up for \(1<p\le 1+ (2+\gamma )/Q\) , while global solutions exist for \(p>1+ (2+\gamma )/(Q + \gamma )\) . In particular, our results coincide with the results found by Georgiev and Palmieri in [17].
