Energy concentration of the focusing energy-critical FNLS

Abstract

We consider the fractional nonlinear Schrödinger equation (FNLS) with general dispersion $|\nabla {|}^{\alpha }$ and focusing energy-critical nonlinearities $-|u{|}^{\frac{2\alpha }{d-\alpha }}u$ and $-(|x{|}^{-2\alpha }\ast |u{|}^2)u$. By adopting Kenig-Tsutsumi \cite{mets}, Kenig-Merle \cite{keme} and Killip-Visan \cite{kv} arguments, we show the energy concentration of radial solutions near the maximal existence time. For this purpose we use Sobolev inequalities for radial functions and establish strong energy decoupling of profiles. And we also show that when the kinetic energy is confined the maximal existence time is finite for some large class of initial data satisfying the initial energy $E(\phi )$ is less than energy of ground state $E(W_{\alpha })$ but $\Vert |\nabla {|}^{\frac{\alpha }{2}}\phi {\Vert }_{L^2}\ge \Vert |\nabla {|}^{\frac{\alpha }{2}}{W}_{\alpha }{\Vert }_{L^2}$.

References in corpus (4)

• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Profile decompositions and Blowup phenomena of mass critical fractional Schrödinger equations
• Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation
• On fractional Choquard equations

Cited by in corpus (1)

• Almost sure local wellposedness of energy critical fractional Schrödinger equations with hartree nonlinearity