REGULARITY AND BLOW-UP CONSTANTS OF SOLUTIONS FOR NONLINEAR DIFFERENTIAL EQUATION

Abstract

In this paper we gain some results on the regularity and also the blow-up rates and constants of solutions to the equation $u'' - u^{p} = 0$ under some different situations. The blow-up rate and blow-up constant of $u^{(2n)}$ are $(p-2n+2)$ and $(\pm) (p-2n+2) \cdot \Pi_{i=0}^{n-1} (p-2i+2) (p-2i+1) E(0)^{p/2}$ respectively; blow-up rate and blow-up constant of $u^{(2n+1)}$ are $(p-2n+1)$ and $(p-2n+2) \Pi_{i=0}^{n-1} (p-2i+2) \cdot (p-2i+1) E(0)^{p-n}$ respectively, where $E(0) = u'(0)^{2} - \frac{2}{p+1} u(0)^{p+1}$.