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In this paper we study the Cauchy problem for the semilinear damped waveequation for the sub-Laplacian on the Heisenberg group. In the case of thepositive mass, we show the global in time well-posedness for small data forpower like nonlinearities. We also obtain similar well-posedness results forthe wave equations for Rockland operators on general graded Lie groups. Inparticular, this includes higher order operators on $\mathbb R^n$ and on theHeisenberg group, such as powers of the Laplacian or of the sub-Laplacian. Inaddition, we establish a new family of Gagliardo-Nirenberg inequalities ongraded Lie groups that play a crucial role in the proof but which are also ofinterest on their own: if $G$ is a graded Lie group of homogeneous dimension$Q$ and $a>0$, $1<r<\frac{Q}{a},$ and $1\leq p\leq q\leq \frac{rQ}{Q-ar},$ thenwe 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 homogeneousSobolev space of order $a$ over $L^r$. If $\frac{a}Q+\frac1p-\frac1r=0$, wehave $p=q=\frac{rQ}{Q-ar}$, and then the above inequality holds for any $0\leqs\leq 1$.

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