Rapheal Oladipo Fifelola; Adedapo Kehinde Femi

Linear Profile Decomposition for the Nonlinear Schrödinger Equation in Exterior Domains

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Author(s)

Rapheal Oladipo Fifelola ^1,* Adedapo Kehinde Femi ²

1. Department of Mathematics, University of Ilorin, Nigeria

2. Department of Physical and Chemical Sciences, Faculty of Science, Federal University of Health Sciences, Ila-Orangun, Osun State, Nigeria

* Corresponding author.

DOI: https://doi.org/10.5815/ijmsc.2026.03.03

Received: 10 Feb. 2026 / Revised: 17 Mar. 2026 / Accepted: 12 Apr. 2026 / Published: 8 Aug. 2026

Index Terms

Profile Decomposition, Concentration-Compactness, Nonlinear Schrödinger Equation, Exterior Domain, Scattering, Strichartz Estimates

Abstract

This paper establishes a linear profile decomposition for bounded sequences in the homogeneous Dirichlet–Sobolev space H ̇_D^1 (Ω), where Ω=R³/O is the exterior of a smooth, compact obstacle O⊂R³. Given a bounded sequence {f_n}⊂H₃¹ (Ω), we prove that, after passing to a subsequence, it decomposes as f_n=∑_(j=1)^Jϕ_n^j +w_n^J, where the profiles {ϕ_n^j} are asymptotically orthogonal and the remainder w_n^J vanishes in all Strichartz spaces L_t^q L_x^r as J→∞. The decomposition satisfies an exact energy identity ∥∇f_n ∥_(L^2)^2=∑_j^∥ ∇ϕ_n^j ∥_(L^2)^2+∥∇w_n^J ∥_(L^2)^2+o(1). Four distinct geometric concentration regimes are identified according to the behaviour of the scale sequence {λ_n^j} relative to the distance d(x_n^j ) to ∂Ω: profiles localised inside Ω; profiles dispersing to spatial infinity (limiting domain R³); profiles concentrating deep inside Ω away from the boundary; and profiles concentrating near ∂Ω (limiting domain: a half-space). As an application, we prove small-data scattering in H ̇_D^1 (Ω) for the defocusing, energy-subcritical NLS i∂_t u+Δ_Ω u=|u|^(p-1) u with 1<p<5 and Dirichlet boundary condition. We emphasise that the NLS is a Hamiltonian (conservative) system: its energy is conserved, not decaying, and the scattering result follows from Strichartz estimates rather than from any dissipative mechanism.

Cite This Paper

Rapheal Oladipo Fifelola, Adedapo Kehinde Femi, "Linear Profile Decomposition for the Nonlinear Schrödinger Equation in Exterior Domains", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.12, No.3, pp. 39-64, 2026. DOI: 10.5815/ijmsc.2026.03.03

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