Scattering of non-radial solutions for the fourth-order Schrödinger equation with Ḣ1/2-critical and supercritical nonlinearity
Abstract
We consider the focusing nonlinear fourth-order Schrödinger equation i ∂t u + μΔu − Δ2 u = − |u|p−1 u, (t,x) ∈ ℝ × ℝd, where μ≥ 0. The scattering of radial solutions below the ground state is shown by Guo (2016) and Dinh (2021) in the case where d ≥ 2 and 1 + 8/d < p < 1 + 8/(d−4)+. In the present paper, we prove the scattering of non-radial solutions below the ground state in the case where d ≥ 2 and 1 + 8/(d−1) ≤ p < 1 + 8/(d−4)+. Our proof is based on the concentration compactness argument by Kenig--Merle (2006). To overcome difficulties in the non-radial case, we use the virial identity in the direction orthogonal to the momentum vector. The condition p ≥ 1 + 8/(d−1) is required to show the positivity of the functional in that virial identity.
Related
- Slow blowup solutions for the focusing energy supercritical nonlinear Schrödinger equation
- Normalized solutions to mass supercritical Schrödinger equations with radial potentials
- Well-posedness for the Cubic Nonlinear Schrödinger Equation With Nonlinearity Supported on a Hypersurface
- Existence of multi-solitons for logarithmic Schrödinger equation with a perturbation of a smooth L2-subcritical function nonlinearity
- Multiple solutions for planar Schrodinger-Poisson system with logarithmic nonlinearity
BibTeX
@misc{inui2026scattering,
title = {{Scattering of non-radial solutions for the fourth-order Schrödinger equation with $\dotH^1/2$-critical and supercritical nonlinearity}},
author = {Takahisa Inui and Koichi Komada and Kuranosuke Nishimura},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05802},
eprint = {2610.05802},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05802},
}
RIS
TY - UNPB
TI - Scattering of non-radial solutions for the fourth-order Schrödinger equation with Ḣ^1/2-critical and supercritical nonlinearity
AU - Takahisa Inui
AU - Koichi Komada
AU - Kuranosuke Nishimura
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05802
UR - https://arxiv.org/abs/2610.05802
AB - We consider the focusing nonlinear fourth-order Schrödinger equation i ∂_t u + μΔu − Δ^2 u = − |u|^p−1 u, (t,x) ∈ ℝ × ℝ^d, where μ≥ 0. The scattering of radial solutions below the ground state is shown by Guo (2016) and Dinh (2021) in the case where d ≥ 2 and 1 + 8/d < p < 1 + 8/(d−4)_+. In the present paper, we prove the scattering of non-radial solutions below the ground state in the case where d ≥ 2 and 1 + 8/(d−1) ≤ p < 1 + 8/(d−4)_+. Our proof is based on the concentration compactness argument by Kenig--Merle (2006). To overcome difficulties in the non-radial case, we use the virial identity in the direction orthogonal to the momentum vector. The condition p ≥ 1 + 8/(d−1) is required to show the positivity of the functional in that virial identity.
ER -