Inhomogeneous parabolic equations with Hardy potential and memory on the Heisenberg group

TL;DR

This paper investigates inhomogeneous parabolic equations on the Heisenberg group with Hardy potentials, nonlocal memory, and forcing, revealing conditions for blow-up, well-posedness, and lifespan estimates through advanced analytical techniques.

Contribution

It introduces new analytical methods to handle the combined effects of Hardy potentials and memory terms on the Heisenberg group, including positivity estimates and blow-up analysis.

Findings

Identified parameter regimes for finite-time and instantaneous blow-up.

Established local well-posedness without Hardy potential.

Derived explicit lifespan estimates under forcing conditions.

Abstract

We study a class of inhomogeneous parabolic equations on the Heisenberg group $\mathbbm{H}^N$ with Hardy-type singular potentials, nonlocal memory terms, and a space-time forcing term: \begin{align} \partial_tu-\Delta_{H}u=\lambda \frac{\psi u}{\|\cdot\|^{2}_{H}}+\frac{1}{\Gamma(\gamma)}\int_0^t(t-\tau)^{\gamma-1}|u(\tau)|^{p}d\tau+t^\alpha f \text{ in } \,\mathbbm{H}^N\times (0,T). \end{align} Here, $\gamma\in [0,1),$ $\alpha\in (-1,\infty),$ $p>1,$ $\lambda>0,$ and $\psi(\cdot)=|\nabla_{H}\|\cdot\|_{H}|^2,$ where $\nabla_H$ is the horizontal gradient associated to $\Delta_H.$ Also, $\|\cdot\|_{H}$ and $\Delta_{H}$ denote the Kor\'anyi norm and sub-Laplacian associated with the sub-Riemannian geometry of $\mathbbm{H}^N,$ respectively. The combination of a singular Hardy potential and a memory kernel introduces significant analytical challenges. Using a Harnack-type inequality adapted to the Heisenberg group setting, we obtain quantitative positivity estimates that enable a detailed blow-up analysis. We identify parameter regimes depending on $p,\gamma,\alpha$ leading to finite-time blow-up or instantaneous blow-up, and establish local well-posedness in the absence of the Hardy potential. These results reveal an interplay between the spatial singularity, temporal nonlocality and a time-dependent forcing term. Finally, under a suitable lower bound on the forcing term $f,$ we derive an explicit lifespan estimate for local-in-time solutions.