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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:265

Article

Ruzhansky, Michael¹  Tokmagambetov, Niyaz^2,3 

[1] Imperial Coll London, Dept Math, 180 Queens Gate, London SW7 2AZ, England

[2] Al Farabi Kazakh Natl Univ, 71 Al Farabi Ave, Alma Ata 050040, Kazakhstan

[3] Inst Math & Math Modeling, 125 Pushkin St, Alma Ata 050010, Kazakhstan

关键词: Nonlinear damped wave equation;    Heisenberg group;    Sub-Laplacian;    Rockland operator;    Graded group;    Gagliardo-Nirenberg inequality;   

DOI  :  10.1016/j.jde.2018.06.033

来源: Elsevier

【 摘 要 】

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 R-n 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 a 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 < Q/a, and 1 <= p < q <= rQ/Q-ar, then we have the following Gagliardo-Nirenberg type inequality parallel to u parallel to(Lq(G)) less than or similar to parallel to u parallel to(s)((L) over dotar(G)) parallel to u parallel to(1-s)(Lp(G)) for s = (1/p - 1/q)(a/Q + 1/p - 1/r)(-1) is an element of [0, 1] provided that a/Q + 1/p - 1/r not equal 0, where (L) over dot(a)(r) is the homogeneous Sobolev space of order a over L-r. If a/Q + 1/p - 1/r= 0, we have p = q = rQ/Q-ar, and then the above inequality holds for any 0 <= s <= 1. (C) 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/).

【 授权许可】

Free