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Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups

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Title: Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups
Authors: Ruzhansky, M
Tokmagambetov, N
Item 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 $0\leqs\leq 1$.
Issue Date: 15-Nov-2018
Date of Acceptance: 26-Jun-2018
URI: http://hdl.handle.net/10044/1/61851
DOI: https://doi.org/10.1016/j.jde.2018.06.033
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/Funder: The Leverhulme Trust
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
The Leverhulme Trust
Funder's Grant Number: RPG-2014-002
EP/K039407/1
EP/R003025/1
RPG-2017-151
Keywords: Science & Technology
Physical Sciences
Mathematics
Nonlinear damped wave equation
Heisenberg group
Sub-Laplacian
Rockland operator
Graded group
Gagliardo-Nirenberg inequality
SPACES
INEQUALITIES
HARDY
math.AP
math.AP
math.FA
35L71, 35L75, 35R03, 22E25
math.AP
math.AP
math.FA
35L71, 35L75, 35R03, 22E25
0101 Pure Mathematics
0102 Applied Mathematics
General Mathematics
Notes: 21 pages
Publication Status: Published
Online Publication Date: 2018-07-03
Appears in Collections:Pure Mathematics
Mathematics
Faculty of Natural Sciences



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