A singularly perturbed boundary-value problem arising in phase transitions

This paper is concerned with the positive solutions of the boundary-value problem

{εu″ - σ(u) = -γ, u(0) = u(1) = 0,

where ε is a small positive parameter and γ is a positive constant. The nonlinear term σ(u) behaves like a cubic; it vanishes only at u = 0, where σ′(0) > 0 and σ″(0) <0. This problem arises in a study of phase transitions in a slender circular cylinder composed of an incompressible phase-transforming material. Here, we determine the number of solutions to the problem for any given γ, derive asymptotic formulas for these solutions, and show that the error terms associated with these formulas are exponentially small, except for one critical value of γ. Our approach is again based on the shooting method used previously by Ou & Wong (Stud. Appl. Math. 112 (2004), 161-200). © 2007 Cambridge University Press.