Slow blowup solutions for the focusing energy supercritical nonlinear Schrödinger equation
Abstract
We consider the focusing energy supercritical nonlinear Schrödinger equation i∂tψ+Δψ+|ψ|2pψ=0 in dimensions d≥11, with p∈ℕ∗, under the Joseph--Lundgren condition 2p+1>pJL(d). For every ν>ν0(d,p), there exists a radial finite-time type II blow-up solution concentrating the ground state Q in the form ψ(t,x)=λ1/p(t)Q(λ(t)x)+ζ(t,x), λ(t)=t−1/2−ν. All homogeneous Sobolev norms |ψ(t)|Ḣs, 1≤ s<sc, remain bounded. The radiation converges to a limiting profile ζ*∈ Ḣs(ℝd), sc<s<sc+2ν|λ+|, where λ+ is determined by the leading correction in the asymptotic expansion of Q at infinity. Our proof combines matched asymptotic expansions across three spatial scales with modified energy estimates. The key ingredient is the systematic treatment of the singular nonlinear background and coupled oscillations generated by the slow algebraic tail of Q.
Related
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BibTeX
@misc{li2026slow,
title = {{Slow blowup solutions for the focusing energy supercritical nonlinear Schrödinger equation}},
author = {Xuanying Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34448},
eprint = {2610.34448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34448},
}
RIS
TY - UNPB
TI - Slow blowup solutions for the focusing energy supercritical nonlinear Schrödinger equation
AU - Xuanying Li
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34448
UR - https://arxiv.org/abs/2610.34448
AB - We consider the focusing energy supercritical nonlinear Schrödinger equation i∂_tψ+Δψ+|ψ|^2pψ=0 in dimensions d≥11, with p∈ℕ^∗, under the Joseph--Lundgren condition 2p+1>p_JL(d). For every ν>ν_0(d,p), there exists a radial finite-time type II blow-up solution concentrating the ground state Q in the form ψ(t,x)=λ^1/p(t)Q(λ(t)x)+ζ(t,x), λ(t)=t^−1/2−ν. All homogeneous Sobolev norms |ψ(t)|_Ḣ^s, 1≤ s<s_c, remain bounded. The radiation converges to a limiting profile ζ^*∈ Ḣ^s(ℝ^d), s_c<s<s_c+2ν|λ_+|, where λ_+ is determined by the leading correction in the asymptotic expansion of Q at infinity. Our proof combines matched asymptotic expansions across three spatial scales with modified energy estimates. The key ingredient is the systematic treatment of the singular nonlinear background and coupled oscillations generated by the slow algebraic tail of Q.
ER -