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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 $$(p(t)u')'=p(t)f(u),\quad u(0)=B,\ u'(0)=0,$$ on the half-line $[0,\infty)$. Here $B<0$ is a parameter, $p(0)=0$ and $p'(t)>0$ on $(0,\infty)$, $f(L)=0$ for some $L>0$ and $xf(x)<0$ if $L_0<x<L$ and $x\ne 0$. 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 $t\to\infty$, which has great importance in applications.

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