BLOW-UP SOLUTIONS TO THE NONLINEAR SECOND ORDER DIFFERENTIAL EQUATION u

Abstract

In this paper we study the following initial value problem for the nonlinear equation, \[ \left\{ \begin{array}[c]{l} u^{\prime\prime}(t)={u(t)}^{p}(c_{1}+c_{2}u^{\prime}(t)^{q}),~~~p,~q\geq 1,\ c_{1}\geq0,c_{2}\geq0,\\ u(0)=u_{0},~u^{\prime}(0)=u_{1}. \end{array} \right. \] We are interested in the properties of solutions of the above problem. We have found blow-up phenomena and obtained some results on blow-up rates, blow-up constants and life-spans.