$\beta$

$h\left(t,0\right)$

$\frac{\partial h}{\partial t}\left(t,0\right)$

$-\infty <\beta <1$

$h\left(t,0\right)\to 0$

$t\to \infty$

$\frac{\partial h}{\partial t}\left(t,0\right)<0$

$\beta =1$

$h\left(t,0\right)=h\left(0,0\right)$

$0\le t\le \infty$

$\frac{\partial h}{\partial t}\left(t,0\right)=0$

$1<\beta <\frac{5}{3}$

$h\left(t,0\right)\to \infty$

$t\to \infty$

$\frac{\partial h}{\partial t}\left(t,0\right)>0$

$\beta =\frac{5}{3}$

$h\left(t,0\right)\to \infty$

$t\to \infty$

Exponentially

$\frac{\partial h}{\partial t}\left(t,0\right)>0$

$\frac{5}{3}<\beta \le 2$

$h\left(t,0\right)\to \infty$

$t\to t_1$

$\frac{\partial h}{\partial t}\left(t,0\right)>0$