Almost Global Existence for the Fractional Schrödinger Equations

Abstract

We study the time of existence of the solutions of the following nonlinear Schrodinger equation (NLS) $$\begin{aligned} \hbox {i}u_t =(-\Delta +m)^su - |u|^2u \end{aligned}$$ on the finite x-interval $$[0,\pi ]$$ with Dirichlet boundary conditions $$\begin{aligned} u(t,0)=0=u(t,\pi ),\qquad -\infty< t<+\infty , \end{aligned}$$ 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 $$0<s<1/2$$ .