Blow-up and global mild solutions for a Hardy-Hénon 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-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 $|\cdot {|}_{\mathbb{H}}^{\gamma }$ is given by the Korányi norm. Our study focuses on nonnegative solutions. We establish that for $\gamma \ge 0$, 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 $\gamma =0$.

26 pages