  It is believed or conjectured that the semilinear wave equations with
scattering space dependent damping admit the Strauss critical exponent, see
Ikehata-Todorova-Yordanov \cite{ITY}(the bottom in page 2) and
Nishihara-Sobajima-Wakasugi \cite{N2}(conjecture iii in page 4). In this work,
we are devoted to showing the conjecture is true at least when the decay rate
of the space dependent variable coefficients before the damping is larger than
2. Also, if the nonlinear term depends only on the derivative of the solution,
we may prove the upper bound of the lifespan is the same as that of the
solution of the corresponding problem without damping. This shows in another
way the \lq\lq hyperbolicity" of the equation.
