Behavior of solutions of a nonlinear ordinary differential equation (Q1380323)

The authors consider positive solutions to initial value problems \[ (\omega^{-\beta} \omega')'- {y\over\beta +2}\omega' +{1\over \beta+2} \omega=0,\;y>0 \tag{*} \] \[ \omega(0) =\eta,\quad \omega' (0)=1, \tag{**} \] where \(0<\beta <\infty\) and \(\eta\) is a positive parameter. It is shown that the existence and nonexistence of global solutions to (*) and (**) strongly depend on \(\eta\) and the asymptotic behaviour of solutions varies according to different choices of \(\beta\). Such problems arise in the study of the quenching rate for solutions to a singular diffusion equation \[ u_t= (u^{-\beta} u_r)_r +{n-1 \over r} u^{-\beta} u_r, \quad 0<r <R,\;t>0, \] \[ u_r(0,t)=0, \quad u_r(R,t)= -1,\;t>0 \] \[ u(r,0)= u_0(r), \quad 0<r <R. \]