Journal of Inequalities and Applications
Volume 5 (2000), Issue 5, Pages 487-496

Singular solutions of a singular differential equation

Takaŝi Kusano1 and Manabu Naito2

1Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan
2Department of Mathematical Sciences, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan

Received 20 June 1999; Revised 22 August 1999

Copyright © 2000 Takaŝi Kusano and Manabu Naito. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An attempt is made to study the problem of existence of singular solutions to singular differential equations of the type (|y|α)+q(t)|y|β=0,() which have never been touched in the literature. Here α and β are positive constants and q(t) is a positive continuous function on [0,). A solution with initial conditions given at t=0 is called singular if it ceases to exist at some finite point T(0,). Remarkably enough, it is observed that the equation () may admit, in addition to a usual blowing-up singular solution, a completely new type of singular solution y(t) with the property that limtT0|y(t)|<andlimtT0|y(t)|=. Such a solution is named a black hole solution in view of its specific behavior at t=T. It is shown in particular that there does exist a situation in which all solutions of () are black hole solutions.