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Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics. (English) Zbl 1186.34014

Summary: This paper investigates singular initial problems

(p(t)u ' ) ' =p(t)f(u),u(0)=B,u ' (0)=0,

on the half-line [0,). Here B<0 is a parameter, p(0)=0 and p ' (t)>0 on (0,), f(L)=0 for some L>0 and xf(x)<0 if L 0 <x<L and x0. The existence of a strictly increasing solution to the problem for which there exists a finite c>0 such that u(c)=L is discussed. This is fundamental for the existence of a strictly increasing solution of the problem having its limit equal to L as t, which has great importance in applications.

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34C11Qualitative theory of solutions of ODE: growth, boundedness