Note on the Uniqueness of a Global Positive Solution to the Second Painleve Equation
Southwest Texas State University, Department of Mathematics
The purpose of this note is to study the uniqueness of solutions to u'' - u3 + (t - c)u = 0, for t ∈ (0, + ∞) with Neumann condition at 0. Assuming a certain condition at infinity, Helfer and Weissler  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 as √t as t approaches infinity.
Second Painleve equation, Neumann condition, Global existence
Guedda, M. (2001). Note on the uniqueness of a global positive solution to the second Painleve equation. <i>Electronic Journal of Differential Equations, 2001</i>(49), pp. 1-4.