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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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NonlinearDampedWaveEquations.pdf | Published version | 395.33 kB | Adobe PDF | View/Open |

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 |