Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 49, pp. 1-4.
Title: Note on the uniqueness of a global positive solution to the second
Painleve equation
Author: Mohammed Guedda (Univ. de Picardie Jules Verne, France)
Abstract:
The purpose of this note is to study the uniqueness of solutions to
$ u'' -u^3 + (t-c)u = 0$, for $ t \in (0,+\infty)$ with
Neumann condition at 0. Assuming a certain conditon at infinity,
Helfer and Weissler [6] have found a unique solution.
We show that, without any assumptions at infinity,
this problem has exactly one global positive solution.
Moreover, the solution behaves like $\sqrt{t}$ as $t$ approaches infinity.
Submitted February 08, 2001. Published July 9, 2001.
Math Subject Classifications: 34B15, 35B05, 82D55.
Key Words: Second Painleve equation; Neumann condition; global existence.