On transitional solutions of second order nonlinear differential equations (Q1766703)

Consider the scalar differential equation \[ u''= f(t,u,u'),\tag{\(*\)} \] where \(f\) is continuous and satisfies \[ f(t,0,0)\equiv 0,\quad f(t,1,0)\equiv 0\quad\text{for }t\in \mathbb{R}. \] The authors ask for a solution of \((*)\) satisfying \[ \lim_{t\to-\infty} u(t)= 0,\quad \lim_{t\to\infty} u(t)= 1,\quad 0\leq u(t)\leq 1\quad\text{for }t\in\mathbb{R}.\tag{\(**\)} \] They derive conditions guaranteeing that the boundary value problem \((*)\) and \((**)\) has at least one solution which is monotone increasing. The proofs are based on comparison principles. The existence of such heteroclinic solutions is related to the existence of traveling wave solutions to partial differential equations.