Regularity and blow-up constants of solutions for nonlinear differential equation (Q2495256)

The authors obtain results concerning regularity of solutions \(u(t)\) of \[ u''-u^p=0,\quad p>1. \] The estimates of blow-up rate and blow-up constant of \(u^{(n)}\) are obtained. It is shown that the blow-up rate and blow-up constant of \(u^{(2n)}\) are \(p-2n+2\) and \[ \pm (p-2n+2)\prod_{i=0}^{n-1} (p-2i+2)(p-2i+1)E^{\frac p2}(0) \] respectively and blow-up rate and blow-up constant of \(u^{(2n+1)}\) are \(p-2n+1\) and \[ (p-2n+2)\prod_{i=0}^{n-1} (p-2i+2)(p-2i+1)E^{p-n}(0) \] respectively, where \(E(0)=(u'(0))^2-\frac{2}{p+1} u^{p+1}(0).\)