Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group
Priyank Oza, Durvudkhan Suragan
TL;DR
This work extends the Fujita-type theory to a nonlocal semilinear heat equation on the Heisenberg group, driven by the fractional sub-Laplacian $(-\Delta_{\mathbbm{H}^N})^s$ with forcing. The authors establish the Fujita threshold $p_F=\frac{Q}{Q-2s}$, where $Q=2N+2$ is the homogeneous dimension, proving global existence for $p>p_F$, finite-time blow-up for $1<p<p_F$, and nonexistence at the critical $p=p_F$ under suitable forcing. The analysis blends fixed-point techniques, heat-kernel estimates for the fractional sub-Laplacian, and the geometry of the Heisenberg group (Korányi distance and homogeneous norms) to transfer Fujita phenomena from Euclidean to sub-Riemannian, nonlocal settings. This advances understanding of nonlinear integro-differential equations in non-Euclidean spaces and highlights the role of sub-Riemannian structure in global behavior of solutions.
Abstract
In this paper, we study the semilinear heat equation with a forcing term, driven by the fractional sub-Laplacian (-Δ_{\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.