Decay of mass for a semilinear heat equation on Heisenberg group

In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation with time-dependent absorption $u_t-{\mathrm{\Delta }}_{\mathbb{H}}u=-k(t)u^p$ posed on ${\mathbb{H}}^n$, driven by the Heisenberg Laplacian and supplemented with a nonnegative integrable initial data, where $p>1$, $n\ge 1$, and $k:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for $p\le 1+2\mathrm{/}Q,$ while the classical/anomalous diffusion effects win if $p>1+2\mathrm{/}Q$, where $Q=2n+2$ is the homogeneous dimension of ${\mathbb{H}}^n$.