Continuous dependence of the Cauchy problem for the inhomogeneous biharmonic NLS equation in Sobolev spaces

2023/05/29 by An, JinMyong, Jo, YuIl, Kim, JinMyong · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35Q55 #Secondary 35B30

paper · doi:10.48550/arxiv.2305.17900

Abstract

In this paper, we study the continuous dependence of 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), in the standard sense in H^s, i.e. in the sense that the local solution flow is continuous H^s→ H^s. Here d∈ \mathbb N, s>0, λ∈ \mathbb R and σ>0. To arrive at this goal, we first obtain the estimates of the term f(u)-f(v) in the fractional Sobolev spaces which generalize the similar results of An-Kim [5](2021) and Dinh [16](2018), where f(u) is a nonlinear function that behaves like λ|u|^σu with λ∈ \mathbb R. These estimates are then applied to obtain the standard continuous dependence result for IBNLS equation with 0

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• Biharmonic non-linear Schrödinger equation with an unbounded inhomogeneous term
• A note on the fourth-order Schrodinger equation with spatially growing inhomogeneous source term

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