Extendable and nonextendable solutions of a nonlinear differential equation of arbitrary order
The paper is devoted to an equation (1) \(u^{(n)}=p(t)| u|^{\lambda}sgn u\), \(t\geq 0\), where p(t)\(\geq 0\) is a piecewise continuous function and \(\lambda >1\). Considering initial conditions (3) \(u^{(i)}(t_ 0)\geq 0\) (or \(u^{(i)}(t_ 0)\leq 0)\), \(t_ 0\geq 0\), \(i=0,1,...,n-1\) and denoting \(J_{\mu}[\phi (\tau)]=\int^{+\infty}_{1}p^{\mu}(\tau)\phi (\tau)d\tau\), the author proves the following three theorems: Theorem 1. If \(J_{\mu}[\tau^{(n-1)\lambda \nu +\nu -1}]=+\infty\), \(\nu <\mu \in (0,1/n)\), then the initial value problem (1), (3) has no proper (unboundedly extendable to the right) solution u(t)\(\not\equiv 0\). Theorem 2. For any \(\mu\geq 1/n\) and \(\phi (t)>0\) there is a function p(t)\(\geq 0\) such that \(J_{\mu}[\phi (\tau)]=+\infty\) and (1) has an n- parametric system of proper solutions with conditions \(0\neq (u(0),u'(0),...,u^{(n-1)}(0))\in \{x: 0\leq x_ i\leq 1\), \(i=1,...,n\}\). Theorem 3. There are p(t)\(\geq 0\) and an unbounded monotonic function \(\phi\) (t)\(\geq 1\) such that \(J_ 1[\tau^{\lambda}/\phi (\tau)]=+\infty\) and the second order equation (1) has a two-parametric system of proper solutions u(t) with the property \(0\leq u'(t)<t^{- 1}\phi (t)u(t)\), \(t>0\). The paper extends previous results of I. T. Kiguradze, G. G. Kvinikadze and of the author.
differential equations of the n-th order
scientific article; zbMATH DE number 3882747