  It is well-known that the critical exponent for semilinear damped wave
equations is Fujita exponent when the damping is effective. Lai, Takamura and
Wakasa in 2017 have obtained a blow-up result not only for super-Fujita
exponent but also for the one closely related to Strauss exponent when the
damping is scaling invariant and its constant is relatively small,which has
been recently extended by Ikeda and Sobajima. Introducing a multiplier for the
time-derivative of the spatial integral of unknown functions, we succeed in
employing the technics on the analysis for semilinear wave equations and
proving a blow-up result for semilinear damped wave equations with sub-Strauss
exponent when the damping is in the scattering range.
