Fujita type results for a parabolic inequality with a non-linear convolution term on the Heisenberg group

The purpose of this paper is to investigate the non-existence of global weak solutions of the following degenerate inequality on the Heisenberg group $$\{\begin{array}{l}{u}_t-{\mathrm{\Delta }}_{\mathbb{H}}u\ge (\mathcal{K}{\ast }_{{}_{\mathbb{H}}}\mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^q,\eta \in {\mathbb{H}}^n,t>0,\\ \\ u(\eta ,0)=u_0(\eta ),\eta \in {\mathbb{H}}^n,\end{array}$$ where $n\ge 1$, $p,q>0$, $u_0\in L_{loc}^1({\mathbb{H}}^n)$, ${\mathrm{\Delta }}_{\mathbb{H}}$ is the Heisenberg Laplacian, and $\mathcal{K}:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ is a continuous function satisfying $\mathcal{K}(\mathrm{\mid }\cdot {\mathrm{\mid }}_{{}_{\mathbb{H}}})\in L_{loc}^1({\mathbb{H}}^n)$ which decreases in a vicinity of infinity. In addition, ${\ast }_{{}_{\mathbb{H}}}$ denotes the convolution operation in ${\mathbb{H}}^n$. Our approach is based on the non-linear capacity method.