  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 $\mathbb R^n$ and on the
Heisenberg group, such as powers of the Laplacian or of 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 $0\leq
s\leq 1$.
