Existence of hypercylinder expanders of the inverse mean curvature flow

We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in $mathbb{R}^ntimes mathbb{R}$, $nge 2$, of the form $(r,y(r))$ or $(r(y),y)$ where $r=|x|$, $xinmathbb{R}^n$, is the radially symmetric coordinate and $yin mathbb{R}$. More precisely for any $lambda>frac{1}{n-1}$ and $mu>0$, we will give a new proof of the existence of a unique even solution $r(y)$ of the equation $frac{r(y)}{1+r(y)^2}=frac{n-1}{r(y)}-frac{1+r(y)^2}{lambda(r(y)-yr(y))}$ in $mathbb{R}$ which satisfies $r(0)=mu$, $r(0)=0$ and $r(y)>yr(y)>0$ for any $yinmathbb{R}$. We will prove that $lim_{ytoinfty}r(y)=infty$ and $a_1:=lim_{ytoinfty}r(y)$ exists with $0le a_1<infty$. We will also give a new proof of the existence of a constant $y_1>0$ such that $r(y_1)=0$, $r(y)>0$ for any $0<y<y_1$ and $r(y)<0$ for any $y>y_1$.