  In this note, we study the Cauchy problem of the semilinear damped wave
equation and our aim is the small data global existence for noncompactly
supported initial data. For this problem, Ikehata and Tanizawa [5] introduced
the energy method with the exponential-type weight function $e^{|x|^2/(1+t)}$,
which is the so-called Ikehata--Todorova--Yordanov type weight. In this note,
we suggest another weight function of the form $(1+|x|^2/(1+t))^{\lambda}$,
which allows us to treat polynomially decaying initial data and give a simpler
proof than the previous studies treating such initial data.
