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It is believed or conjectured that the semilinear wave equations withscattering space dependent damping admit the Strauss critical exponent, seeIkehata-Todorova-Yordanov \cite{ITY}(the bottom in page 2) andNishihara-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 rateof the space dependent variable coefficients before the damping is larger than2. 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 thesolution of the corresponding problem without damping. This shows in anotherway the \lq\lq hyperbolicity" of the equation.

Strauss exponent for semilinear wave equations with scattering space dependent damping