Nonscattering solutions to the $L^2$-supercritical NLS Equations

Abstract

We investigate the nonlinear Schrödinger equation $i{u}_t+\Delta u+|u{|}^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1,2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of infinite-variance and nonradial solutions. If $M(u{)}^{\frac{1-s_c}{s_c}}E(u)<M(Q{)}^{\frac{1-s_c}{s_c}}E(Q)$ and $\Vert u_0{\Vert }_2^{\frac{1-s_c}{s_c}}\Vert \nabla u_0{\Vert }_2>\Vert Q{\Vert }_2^{\frac{1-s_c}{s_c}}\Vert \nabla Q{\Vert }_2,$ then either $u(t)$~blows up in finite forward time, or $u(t)$ exists globally for positive time and there exists a time sequence $t_n\to +\infty$ such that $\Vert \nabla u(t_n){\Vert }_2\to +\infty .$ Here $Q$ is the ground state solution of $-Q+\Delta Q+|Q{|}^{p-1}Q=0.$ A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation in \cite{holmer10} to the general mass-supercritical and energy-subcritical case .

References in corpus (1)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation