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

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.