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-Δ_{H}u=λ\frac{ψu}{\|\cdot\|^{2}_{H}}+\frac{1}{Γ(γ)}\int_0^t(t-τ)^{γ-1}|u(τ)|^{p}dτ+t^αf \text{ in } \,\mathbbm{H}^N\times (0,T). \end{align} Here, $\gamma \in [0,1),$ $\alpha \in (-1,\infty ),$ $p>1,$ $\lambda >0,$ and $\psi (\cdot )=|{\nabla }_H\Vert \cdot {\Vert }_H{|}^2,$ where ${\nabla }_H$ is the horizontal gradient associated to ${\Delta }_H.$ Also, $\Vert \cdot {\Vert }_H$ and ${\Delta }_H$ denote the Korányi 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.