Decay of mass for a semilinear heat equation on Heisenberg group
Ahmad Z. Fino
TL;DR
This work analyzes the large-time behavior of nonnegative solutions to the semilinear heat equation with time-dependent absorption $u_t-\Delta_{\mathbb{H}}u=-k(t)u^p$ on the Heisenberg group $\mathbb{H}^n$. Using semigroup (heat kernel) estimates, scaling, and nonlinear capacity techniques, the authors establish a Fujita-type dichotomy governed by the homogeneous dimension $Q=2n+2$: the total mass $M_{\mathbb{H}}(t)$ persists ($M_{\mathbb{H}}(t)\to M_\infty>0$) when $p>1+\frac{2}{Q}$ and decays to zero ($M_{\mathbb{H}}(t)\to 0$) when $1<p\le 1+\frac{2}{Q}$. In the former regime, the solution asymptotically resembles a multiple of the heat kernel, $u(t,\cdot)\approx M_\infty h_t$, with quantitative $L^q$ convergence, while in the latter regime, nonlinear absorption dominates and kills mass. The results extend classical Euclidean Fujita-type thresholds to the subelliptic Heisenberg setting and illustrate the interplay between diffusion and nonlinear absorption in a noncommutative geometric context, leveraging $L^p-L^q$ estimates, comparison principles, and nonlinear capacity methods.
Abstract
In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation with time-dependent absorption $u_{t}-Δ_{\mathbb{H}}u=- k(t)u^p$ posed on $\mathbb{H}^n$, driven by the Heisenberg Laplacian and supplemented with a nonnegative integrable initial data, where $p>1$, $n\geq 1$, and $k:(0,\infty)\to(0,\infty)$ 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 $p\leq 1+2/Q,$ while the classical/anomalous diffusion effects win if $p>1+{2}/{Q}$, where $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$.