Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg
group and for Rockland operators on graded Lie groups
group and for Rockland operators on graded Lie groups
File(s)
Published version
Author(s)
Ruzhansky, Michael
Tokmagambetov, Niyaz
Type
Journal Article
Abstract
In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. In particular, this includes higher order operators on and on the Heisenberg group, such as powers of the Laplacian or the sub-Laplacian. In addition, we establish a new family of Gagliardo-Nirenberg inequalities on graded Lie groups that play a crucial role in the proof but which are also of interest on their own: if $G$ is a graded Lie group of homogeneous dimension $Q$ and $a>0$, $1<r<\frac{Q}{a},$ and $1\leq p\leq q\leq \frac{rQ}{Q-ar},$ then we have the following Gagliardo-Nirenberg type inequality $$\|u\|_{L^{q}(G)}\lesssim \|u\|_{\dot{L}_{a}^{r}(G)}^{s} \|u\|_{L^{p}(G)}^{1-s}$$ for $s=\left(\frac1p-\frac1q\right)
\left(\frac{a}Q+\frac1p-\frac1r\right)^{-1}\in [0,1]$ provided that
$\frac{a}Q+\frac1p-\frac1r\not=0$, where $\dot{L}_{a}^{r}$ is the homogeneous Sobolev space of order $a$ over $L^r$. If $\frac{a}Q+\frac1p-\frac1r=0$, we have $p=q=\frac{rQ}{Q-ar}$, and then the above inequality holds for any $0\leqs\leq 1$.
\left(\frac{a}Q+\frac1p-\frac1r\right)^{-1}\in [0,1]$ provided that
$\frac{a}Q+\frac1p-\frac1r\not=0$, where $\dot{L}_{a}^{r}$ is the homogeneous Sobolev space of order $a$ over $L^r$. If $\frac{a}Q+\frac1p-\frac1r=0$, we have $p=q=\frac{rQ}{Q-ar}$, and then the above inequality holds for any $0\leqs\leq 1$.
Date Issued
2018-11-15
Date Acceptance
2018-06-26
Citation
Journal of Differential Equations, 2018, 265 (10), pp.5212-5236
ISSN
0022-0396
Publisher
Elsevier
Start Page
5212
End Page
5236
Journal / Book Title
Journal of Differential Equations
Volume
265
Issue
10
Copyright Statement
© 2018 The Authors. Published by Elsevier Inc. This is an open access article under the CC-BY license (http://creativecommons.org/licenses/by/4.0/)
Sponsor
The Leverhulme Trust
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
The Leverhulme Trust
Identifier
http://arxiv.org/abs/1703.07902v1
Grant Number
RPG-2014-002
EP/K039407/1
EP/R003025/1
RPG-2017-151
Subjects
math.AP
math.AP
math.FA
35L71, 35L75, 35R03, 22E25
Notes
21 pages
Publication Status
Published
Date Publish Online
2018-07-03