ON THE LIFE SPAN OF THE SCHRODINGER EQUATION WITH SUB-CRITICAL POWER NONLINEARITY

We discuss the life span of the Cauchy problem for the one-dimensional Schrodinger equation with a single power nonlinearity lambda vertical bar u vertical bar(p-1) (lambda is an element of C, 2 <= p < 3) and initial data of the form epsilon phi prescribed. Here, E stands for the size of the data. It is not difficult to see that the life span T(epsilon) is estimated by C(0)epsilon(-2(p- 1)/(3-p)) from below, provided epsilon is sufficiently small. In this paper, we consider a more precise estimate for T(epsilon) and we prove that hill inf(epsilon-0) epsilon(2(p-1)/(3- p)) T(epsilon) is larger than some positive constant expressed only by p Im lambda and phi.