Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with general nonlinearity in the radial case

2024/06/17 by Jun Wang, Zhaoyang Yin, Wang, Jun +1
Mathematics · #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2406.11532

Abstract

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schrödinger equation with general nonlinearity \i ∂_t u+Δu + f(u)=0, (x, t) ∈ ℝ^N × ℝ,
.u|_t=0=u₀ ∈ H ¹(ℝ^N),. where f:ℂ→ ℂ satisfies Sobolev critical growth condition. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. This paper generalizes the conclusions in \citeKCEMF2006(Invent. Math).

Related

• Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with indefinite potential in the radial case
• Global well-posedness and scattering of the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation with radial data
• Global Well-posedness and Scattering for the Focusing Energy-critical Inhomogeneous Nonlinear Schrödinger Equation with Non-radial Data
• Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in \R¹⁺⁴
• Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data