  The paper investigates the existence of global attractors and their upper
semicontinuity for a structural damped wave equation on $\mathbb{R}^{N}:
u_{tt}-\Delta u+(-\Delta)^\alpha u_{t}+u_{t}+u+g(u)=f(x)$, where $\alpha\in
(1/2, 1)$ is called a dissipative index. We propose a new method based on the
harmonic analysis technique and the commutator estimate to exploit the
dissipative effect of the structural damping $(-\Delta)^\alpha u_{t}$ and to
overcome the essential difficulty: "both the unbounded domain $\mathbb{R}^N$
and the supercritical nonlinearity cause that the Sobolev embedding loses its
compactness"; Meanwhile we show that there exists a supercritical index
$p_\alpha\equiv\frac{N+4\alpha}{N-4\alpha}$ depending on $\alpha$ such that
when the growth exponent $p$ of the nonlinearity $g(u)$ is up to the
supercritical range: $1\leqslant p<p_\alpha$: (i) the IVP of the equation is
well-posed and its solution is of additionally global smoothness when $t>0$;
(ii) the related solution semigroup possesses a global attractor
$\mathcal{A}_\alpha$ in natural energy space for each $\alpha\in (1/2, 1)$;
(iii) the family of global attractors $\{\mathcal{A}_\alpha\}_{\alpha\in (1/2,
1) }$ is upper semicontinuous at each point $\alpha_0\in (1/2, 1)$.
