Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

2022/08/18 by JinMyong An, An, JinMyong, PyongJo Ryu +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2208.08657

Abstract

We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation iu_t +Δ² u=λ|x|^-b|u|^σu, u(0)=u₀ ∈ H^s (\mathbb R^d), where λ∈ \mathbb R, d∈ \mathbb N, 0≤ s&lt;min\2+(d)/(2),d\, 0<b>0. Applying this estimate and the contraction mapping principle based on Strichartz estimates in Lorentz spaces, we then establish the local well-posedness in H^s for the IBNLS equation in both of subcritical case σ&lt;σ_c(s) and critical case σ=σ_c(s). We also prove that the IBNLS equation is globally well-posed in H^s, if the initial data is sufficiently small and (8-2b)/(d)≤ σ≤ σ_c(s) with σ&lt;∞.</b>

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