Generalized painlevé equation and superconductivity. asymptotic behavior of unbounded solutions

Abstract

Consider the nonlinear scalar differential equation 1 p(t) (p(t)y′(t))′=−(t)f(t,y(t),p(t)y′(t)) , where p and q are positive on (0, 1), “singular” at t = 0. 1 and/or y = 0 and f ϵ C( R + x R + x R +, R +), associated to the boundary conditions x(0)=a⩾0, lim t→∞ p(t)x′(t)=b⩾0 . We prove the existence of a global, positive and strictly increasing solution x = x( t) of this BVP, such that its “derivative” y = p( t) x( t) is also a positive and strictly decreasing map, under a natural growth in f of “superlinear” type. Our approach is based on the analysis of the corresponding vector field on the face-plane ( x, px′) and the well-known Knesser's type (continuum) technique. As an application, we study the generalized Painlevé equation in a semi-infinite interval (0, +∞) x″= x 2n+1 − (t−c) 2k+1x 2(n−k)−1 which, in turn, models a superheating field attached to a semi-infinite superconductor. Namely, we prove the existence of a (global) strictly positive solution satisfying x′(0)=0, lim t→∞ x(t) t β = M, and lim t→∞ t βx′ (t) = βM , where 0 < β < 1, M > 0, and n, k ϵ N with k < n are arbitrary constants.