Blow-up and global mild solutions for a Hardy-Hénon parabolic equation on the Heisenberg group
Ricardo Castillo, Ricardo Freire, Miguel Loayza
TL;DR
This work analyzes the Hardy–Hénon parabolic equation $u_t - \Delta_{\mathbb{H}} u = |\cdot|_{\mathbb{H}}^{\gamma} u^p$ on the Heisenberg group $\mathbb{H}^N$, focusing on nonnegative mild solutions and determining Fujita-type thresholds in terms of the homogeneous dimension $Q=2N+2$. Using semigroup estimates for the sub-Laplacian $\Delta_{\mathbb{H}}$ and weighted function spaces, it derives sharp blow-up and global existence criteria: for $\gamma \ge 0$ the critical exponent is $p_c = 1+ (2+\gamma)/Q$, with blow-up for $1<p\le p_c$ and global small-data solutions for $p>p_c$; for $\gamma<0$ blow-up occurs for $1<p\le p_c$ while global solutions exist for $p>1+(2+\gamma)/(Q+\gamma)$ under suitable initial data. The results extend the Euclidean and Carnot-group literature (recovering Georgiev–Palmieri when $\gamma=0$) and employ a blend of fixed-point, semigroup, and differential-inequality techniques tailored to the Heisenberg setting. The findings contribute precise thresholds and constructive conditions for global behavior vs finite-time blow-up of parabolic equations on sub-Riemannian manifolds.
Abstract
We are concerned with the existence of global and blow-up solutions for the nonlinear parabolic problem described by the Hardy-Hénon equation $u_t - Δ_{\mathbb{H}} u = |\cdot|_{\mathbb{H}}^γ u^p \mbox{ in } \mathbb{H}^N \times (0,T),$ where $\mathbb{H}^N$ is the $N$-dimensional Heisenberg group, and the singular term $|\cdot|_{\mathbb{H}}^γ$ is given by the Korányi norm. Our study focuses on nonnegative solutions. We establish that for $γ\geq 0$, the Fujita critical exponent is $p_c = 1+ (2+γ)/Q$, where $Q=2N+2$ is the homogeneous dimension of $\mathbb{H}^N$. For $γ<0$, the solutions blow up for $1<p<1+ (2+γ)/Q$, while global solutions exist for $p>1+ (2+γ)/(Q + γ)$. In particular, our results coincide with the results found by Georgiev and Palmieri in \cite{PALMIERI} for $γ=0$.