Energy concentration of the focusing energy-critical FNLS

2015/01/31 by Cho, Yonggeun, Hwang, Gyeongha, Shim, Yong-Sun
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1502.00100

Abstract

We consider the fractional nonlinear Schrödinger equation (FNLS) with general dispersion |∇|^α and focusing energy-critical nonlinearities -|u|^(2α)/(d-α)u and -(|x|^-2α * |u|²) u. By adopting Kenig-Tsutsumi \citemets, Kenig-Merle \citekeme and Killip-Visan \citekv 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(φ) is less than energy of ground state E(W_α) but ‖|∇|^\fracα2 φ‖_L² ≥ ‖|∇|^\fracα2 W_α‖_L².

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