Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces: the critical case

TL;DR

This paper investigates the critical case of semilinear hypoelliptic damped wave equations on graded Lie groups, analyzing existence, diffusion phenomena, and initial data in negative Sobolev spaces, extending known results to more general settings.

Contribution

It introduces new results on the critical exponent case for higher-order hypoelliptic damped wave equations on graded Lie groups, including diffusion phenomena and initial data in negative Sobolev spaces.

Findings

Established critical exponent for well-posedness.

Analyzed diffusion phenomena in hypoelliptic settings.

Extended results to higher-order operators on Lie groups and Euclidean spaces.

Abstract

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $\nu\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-\gamma}(\mathbb G), \gamma>0,$ for the critical exponent case $p=1+\frac{2\nu}{Q+2\gamma}.$ We also explore the diffusion phenomenon of the higher-order hypoelliptic damped wave equations on graded Lie groups with initial data belonging to Sobolev spaces of negative order. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups.