Fractional semilinear damped wave equation on the Heisenberg group
Aparajita Dasgupta, Shyam Swarup Mondal, Abhilash Tushir
TL;DR
The paper addresses the Cauchy problem for a semilinear damped wave equation with a fractional sub-Laplacian $(-\\mathcal{L}_{\\mathbb{H}})^{\\alpha}$ on the Heisenberg group $\\mathbb{H}^{n}$, incorporating damping $b$ and mass $m$. It develops explicit linear decay estimates for solutions and their derivatives, distinguishing massive and massless scenarios, and leverages these to prove global well-posedness for small data in several regimes, including $m=0$ with $L^{1}\cap L^{2}$ or $L^{2}$ data and $m>0$ with $1<p\le 1+\frac{2\\alpha}{\\mathcal{Q}-2\\alpha}$. The analysis uses Fourier methods on $\\mathbb{H}^{n}$, the Hermite framework for fractional powers of the sub-Laplacian, and Gagliardo–Nirenberg inequalities adapted to the Heisenberg setting; an application to a weakly coupled two-equation system is also presented. Overall, the work extends decay and global existence results to the fractional-damping context on the Heisenberg group, highlighting the interplay between the homogeneous dimension $\\mathcal{Q}$, the fractional order $\\alpha$, and the nonlinear exponent $p$ in non-Euclidean geometry.
Abstract
This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian $(-\mathcal{L}_{\mathbb{H}})^α$, $α>0$ on the Heisenberg group $\mathbb{H}^{n}$ with power type non-linearity. With the presence of a positive damping term and nonnegative mass term, we derive $L^2-L^2$ decay estimates for the solution of the homogeneous linear fractional damped wave equation on $\mathbb{H}^{n}$, for its time derivative, and for its space derivatives. We also discuss how these estimates can be improved when we consider additional $L^1$-regularity for the Cauchy data in the absence of the mass term. Also, in the absence of mass term, we prove the global well-posedness for $2\leq p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}}$ $(\text{or }1+\frac{4α}{\mathcal{Q}}<p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}})$ in the case of $L^1\cap L^2$ $(\text{or } L^2)$ Cauchy data, respectively. However, in the presence of the mass term, the global (in time) well-posedness for small data holds for $1<p \leq 1+ \frac{2α}{(\mathcal{Q}-2α)_{+}}$. Finally, as an application of the linear decay estimates, we investigate well-posedness for the Cauchy problem for a weakly coupled system with two semilinear fractional damped wave equations with positive mass term on $\mathbb{H}^{n}$.