On the existence and properties of three types of solutions of singular IVPs

Abstract

The paper studies the singular initial value problem $$ (p(t)u'(t))' + q(t)f(u(t))=0, \qquad t>0, \qquad u(0)=u_0\in [L_0,L], \qquad u'(0)=0. $$ Here, $f\in C(\mathbb{R})$, $f(L_0)=f(0)=f(L)=0$, $L_0 0$ for $x\in (L_0,0) \cup (0,L)$. Further, $p,q\in C[0,\infty)$ are positive on $(0,\infty)$ and $p(0)=0$. The integral $\int_0^1 \frac{\mathrm{d}{s}}{p(s)}$ may be divergent which yields the time singularity at $t=0$. The paper describes a set of all solutions of the given problem. Existence results and properties of oscillatory solutions and increasing solutions are derived. By means of these results, the existence of an increasing solution with $u(\infty)=L$ (a homoclinic solution) playing an important role in applications, is proved.

Keywords

• second order ode; time singularity; asymptotic properties; damped oscillatory solution; escape solution; homoclinic solution; unbounded domain.