Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group

TL;DR

This paper determines the critical Fujita exponent for a fractional sub-Laplacian heat equation on the Heisenberg group, establishing conditions for global existence or finite-time blow-up of solutions.

Contribution

It extends the classical Fujita phenomenon to a sub-Riemannian setting with fractional nonlocal operators on the Heisenberg group.

Findings

Fujita exponent is Q/(Q-2s) for the fractional sub-Laplacian on the Heisenberg group.

Global solutions exist for p > p_F, non-existence for 1 < p < p_F.

Finite-time blow-up occurs at the critical exponent p = p_F.

Abstract

In this paper, we study the semilinear heat equation with a forcing term, driven by the fractional sub-Laplacian (-\Delta_{\mathbbm{H}^N})^s of order $s\in (0,1),$ on the Heisenberg group $\mathbbm{H}^N$. We establish that the Fujita exponent, a critical threshold that delimits different dynamical regimes of this equation, is $$p_F\coloneqq\frac{Q}{Q-2s},$$ where $Q\coloneqq 2N+2$ is the homogeneous dimension of $\mathbbm{H}^N$. We prove the existence of global-in-time solutions for the supercritical case $(p>p_F),$ and the non-existence of global-in-time solutions for the subcritical case $(1<p<p_F).$ For the critical case $p=p_F,$ we provide a class of functions for which the solution blows up in finite time. These results extend the classical Fujita phenomenon to a sub-Riemannian setting with the nonlocal effects of the fractional sub-Laplacian. Our proof methods intertwine analytic techniques with the geometric structure of the Heisenberg group.