A note on a nonautonomous O.D.E. related to the Fisher equation
Consider the nonautonomous differential equation \[ d^2u/dt^2 + c \;du/dt + a(t)f(u)=0, \tag \(*\) \] with \(c>0\) under the assumptions: (i) \(f: \mathbb{R}^+ \rightarrow \mathbb{R}^+\) is locally Lipschitzian and satisfies \(f(0)=0.\) (ii) \(a: \mathbb{R} \rightarrow \mathbb{R}\) is continuous. There is \(\alpha > 0\) such that \(\alpha < a (t) \leq 1\) for all \(t\). (iii) \(f(1)=0\), \(f(u) > 0\) for \(0 < u < 1.\) Under these assumptions, the author proves that \((*)\) has a positive solution \(u(t)\) satisfying \(u(- \infty ) =1\), \(u(+\alpha ) =0\), provided \(c\) satisfies some inequality. He establishes this result by comparing \((*)\) with some linear equations. By the same technique, the author proves the existence of a positive solution to \((*)\) satifying \(u(0)=u(\infty) =0.\)
scientific article; zbMATH DE number 1398405