Blow-up and global mild solutions for a Hardy-H\'enon parabolic equation on the Heisenberg group

Abstract

We are concerned with the existence of global and blow-up solutions for the nonlinear parabolic problem described by the Hardy-H\'enon equation $u_t - \Delta_{\mathbb{H}} u = |\cdot|_{\mathbb{H}}^{\gamma} u^p \mbox{ in } \mathbb{H}^N \times (0,T),$ where $\mathbb{H}^N$ is the $N$-dimensional Heisenberg group, and the singular term $|\cdot|_{\mathbb{H}}^{\gamma}$ is given by the Kor\'anyi norm. Our study focuses on nonnegative solutions. We establish that for $\gamma\geq 0$, the Fujita critical exponent is $p_c = 1+ (2+\gamma)/Q$, where $Q=2N+2$ is the homogeneous dimension of $\mathbb{H}^N$. For $\gamma<0$, the solutions blow up for $1<p<1+ (2+\gamma)/Q$, while global solutions exist for $p>1+ (2+\gamma)/(Q + \gamma )$. In particular, our results coincide with the results found by Georgiev and Palmieri in \cite{PALMIERI} for $\gamma=0$.