Almost Global Existence for the Fractional Schrodinger Equations

We study the time of existence of the solutions of the following nonlinear Schrodinger equation (NLS) iu(t) = (-Delta + m)(s)u - vertical bar u vertical bar(2)u on the finite x-interval [0, pi] with Dirichlet boundary conditions u(t, 0) = 0 = u(t, pi), -infinity < t < +infinity, where (-Delta + m)(s) stands for the spectrally defined fractional Laplacian with 0 < s < 1/2. We prove an almost global existence result for the above fractional Schrodinger equation, which generalizes the result in Bambusi and Sire (Dyn PDE 10(2):171-176, 2013) from s > 1/2 to 0 < s < 1/2.