Revisit on global existence of solutions for semilinear damped wave equations in $\mathbb{R}^N$ with noncompactly supported initial data

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.