Asymptotic Expansion of the L^2 Norms of the Solutions to the Heat and Dissipative Wave Equations on the Heisenberg Group

Abstract

Motivated by the recent work on asymptotic expansions of heat and dissipative wave equations on the Euclidean space, and the resurgent interests in Heisenberg groups, this dissertation is devoted to the asymptotic expansions of heat and dissipative wave equations on Heisenberg groups. The Heisenberg group, $\mathbb{H}^{n}$, is the $\mathbb{R}^{2n+1}$ manifold endowed with the law $$(x,y,s)\cdot (x',y',s') = (x+x', y+y', s+ s' + \frac{1}{2} (xy' - x'y)),$$ where $x,y\in \mathbb{R}^{n}$ and $t\in \mathbb{R}$. Let $v(t,z)$ and $u(t,z)$ be solutions of the heat equation, $v_{t} - \mathcal{L} v=0$, and dissipative wave equation, $u_{tt}+u_{t} - \mathcal{L}u =0$, over the Heisenberg group respectively, where $\mathcal{L}$ is the sub-Laplacian. To overcome the Heisenberg group setting, we first establish the Group Fourier transform for an integrable function on the space. The Fourier transform together with the Plancherel formula, help us to obtain the following expansions for $\|u(t,z)\|_{L^{2}(\mathbb{H})}$ and $\|v(t,z)\|_{L^{2}(\mathbb{H})}$ as $t\rightarrow \infty$, $$ \|u(t,\cdot)\|_{L^{2}(\mathbb{H})} \sim \sum\limits_{n=0}^{N-1} b_{n}t^{-n-2} + O(t^{-N-2}), \hspace{10mm} \| v(t,\cdot)\|_{L^{2}(\mathbb{H})} \sim \sum\limits_{n=0}^{N-1}c_{n} t^{-n-2} +O(t^{-N-1}), $$ where $b_{n}$ and $c_{n}$ only depend on the initial conditions.