On the GWP of focusing energy-ciritical inhomogeneous NLS

Abstract

We consider the focussing energy-critical inhomogeneous nonlinear Schrödinger equation: $$i{u}_t+\Delta u+g|u{|}^{2}u=0,u(0)=\phi \in {\dot{H}}^1,0\le g_i\le |x|g\le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(φ) < E_g(Q),\;\;\mbox{and}\;\; g_s\|φ\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $\Delta Q+|x{|}^{-1}{Q}^3=0$, together with scaling condition $|g(x)|+|x||\nabla g(x)|\lesssim |x{|}^{-1}$, variational condition $g_s(2-g_i)\le 1$, and rigidity condition $-g(x)\le x\cdot \nabla g(x)$. We also provide sharp finite time blowup results for nonradial and radial solutions. For this we utilize the localized virial identity.

References in corpus (3)

• Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities
• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Small data scattering of the inhomogeneous cubic-quintic NLS in 2 dimensions