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

Michael Ruzhansky,Niyaz Tokmagambetov

doi:10.1016/j.jde.2018.06.033

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 Rn 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<Qa, and 1≤p≤q≤rQQ−ar, then we have the following Gagliardo–Nirenberg type inequality‖u‖Lq(G)≲‖u‖L˙ar(G)s‖u‖Lp(G)1−s for s=(1p−1q)(aQ+1p−1r)−1∈[0,1] provided that aQ+1p−1r≠0, where L˙ar is the homogeneous Sobolev space of order a over Lr. If aQ+1p−1r=0, we have p=q=rQQ−ar, and then the above inequality holds for any 0≤s≤1.

Highlights

In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub
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 of interest on their own: if G is a graded Lie group of homogeneous dimension
✩ The first author was supported in parts by the EPSRC grants EP/K039407/1 and EP/R003025/1, and by the Leverhulme Grants RPG-2014-02 and RPG-2017-151

Summary

Linear damped wave equation on the Heisenberg group

We will need some elements of the analysis on the Heisenberg group Hn. ∂t u(0, λ) = u1(λ), where σL(λ) is the symbol of −L It takes the form n σL(λ) = |λ|Hw ≡ |λ| (−∂w2 j + wj2), j =1. By solving the second order ordinary differential equation (2.9) with constant coefficients, we get the estimates. There exists a positive constant δ > 0 such that in all the cases we have. The same estimates work if we multiply the equation (2.9) by powers of the spectral decomposition of the symbol of the sub-Laplacian. The Sobolev spaces HLs , s ∈ R, associated to L, is defined as HLs (Hn) := f ∈ D (Hn) : (I − L)s/2f ∈ L2(Hn) , with the norm f HLs (Hn) := (I − L)s/2f L2(Hn).

Semilinear damped wave equations on the Heisenberg group

Nonlinear damped wave equations on graded Lie groups

Gagliardo–Nirenberg inequalities

Linear equation

Semilinear equations

Nonlinear equations