  The work is devoted to Dirichlet problem for sub-quintic semi-linear wave
equation with damping damping term of the form $(-\Delta)^\alpha\partial_t u$,
$\alpha\in(0,\frac{1}{2})$, in bounded smooth domains of $\Bbb R^3$. It appears
that to prove well-posedness and develop smooth attractor theory for the
problem we need additional regularity of the solutions, which does not follow
from the energy estimate. Considering the original problem as perturbation of
the linear one the task is reduced to derivation of Strichartz type estimate
for the linear wave equation with fractional damping, which is the main feature
of the work. Existence of smooth exponential attractor for the natural
dynamical system associated with the problem is also established.
