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

TL;DR

This paper establishes Fujita-type non-existence results for a nonlinear parabolic inequality with a convolution term on the Heisenberg group, extending classical blow-up results to a non-commutative setting.

Contribution

It introduces new non-existence criteria for solutions of a degenerate inequality involving convolution on the Heisenberg group, using the nonlinear capacity method.

Findings

Non-existence of global weak solutions under certain conditions.

Extension of Fujita-type results to the Heisenberg group.

Application of the nonlinear capacity method to convolution inequalities.

Abstract

The purpose of this paper is to investigate the non-existence of global weak solutions of the following degenerate inequality on the Heisenberg group $$ \begin{cases} u_{t}-\Delta_{\mathbb{H}}u\geq (\mathcal{K}\ast_{_{\mathbb{H}}}|u|^p)|u|^q ,\qquad {\eta\in \mathbb{H}^n,\,\,\,t>0,} \\{}\\ u(\eta,0)=u_{0}(\eta), \qquad\qquad\qquad\quad \eta\in \mathbb{H}^n, \end{cases} $$ where $n\geq1$, $p,q>0$, $u_0\in L^1_{loc}(\mathbb{H}^n)$, $\Delta_{\mathbb{H}}$ is the Heisenberg Laplacian, and $\mathcal{K}:(0,\infty)\rightarrow(0,\infty)$ is a continuous function satisfying $\mathcal{K}(|\cdotp|_{_{\mathbb{H}}})\in L^1_{loc}(\mathbb{H}^n)$ 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.