Asymptotic behavior of positive solutions of $x"=-t^{\alpha \lambda -2}x^{1+\alpha}$ with $\alpha <0$ and $\lambda <-1$ or $\lambda >0$

Abstract

In this paper, we consider an initial value problem of the differential equation written in the title under an initial condition $x(T)=A$, $x'(T)=B$ $(0<T<\infty$, $0<A<\infty$, $-\infty <B<\infty)$. In the case $\lambda >0$, we conclude that if $T$ and $A$ are fixed arbitrarily, then there exists a number $B_{1}$ such that in every case of $B = B_{1}$, $B>B_{1}$, and $B<B_{1}$, we get analytical expressions of the solution of the initial value problem valid in the neighborhoods of the ends of the domain of the solution. Moreover we treat the case $\lambda <-1$. This case connects with the boundary layer theory of viscous fluids. The conclusions of this case are got directly from those of the case $\lambda >0$. Finally we discuss the case $T=0$ and $\lambda <-1$.