On existence of singular solutions of \(n\)-th order differential equations. (English) Zbl 1090.34536

Bartušek, Miroslav

Arch. Math., Brno 36, Suppl., 395-404 (2000).

The author considers the \(n\)-th order differential equation \[ y^{(n)}=f(t,y,y',\dots ,y^{(n-2)})g(y^{(n-1)}) (1) \] where \(n \geq 2\), \(f \in C^0([0,\infty ) \times \mathbb {R}^{n-1})\), \(g \in C^0(\mathbb R)\), \(\alpha f(t,x_1,\dots ,x_{n-1})x_1>0\) for \(x_1 \neq 0\) with \(\alpha \in \{-1,1\}\) and \(g(x) \geq 0\) for \(x \in \mathbb R\). A solution \(y(t)\) of (1) defined on \([T,\tau ) \subset [0,\infty )\) is called singular if \(\tau < \infty \) and \(y(t)\) cannot be defined for \(t=\tau \). The paper presents sufficient conditions under which (1) has a singular solution \(y:[T,\tau ) \to \mathbb R\) satisfying \(\lim _{t \to \tau _-}y^{(i)}(t)=c_i \in \mathbb R\), \(i=0,1,\dots ,n-2\) and \(\lim _{t \to \tau _-}| y^{(n-1)}(t)| =\infty \).

Reviewer: Svatoslav Staněk (Olomouc)

MSC:

34C11

Growth and boundedness of solutions to ordinary differential equations

Keywords:

singular solution; black hole solution