Semilinear Schrödinger equation with harmonic potential and combined power-type nonlinearities
Abstract
We study the Cauchy problem for the semilinear Schrödinger equation with a harmonic potential and combined power-type nonlinearities. The goal is to classify initial data that lead to either global existence or blowup solutions by using an enhanced cross-constrained variational theory. First, we review and analyze this variational approach, pinpointing the main issue for our specific problem. We then introduce new variational structures by carefully selecting terms in the energy functional and adjusting their weights, enabling us to create eight cross-constrained variational combinations and related manifolds to distinguish initial data that result in global solutions or finite-time blowup. Using these structures, we examine three initial energy levels: negative, non-negative low, and high energy. Notably, we prove that for any initial data with negative energy, solutions blow up in finite time regardless of their size. Throughout the study, we utilize these new variational frameworks and functionals to systematically classify initial data for global existence or finite-time blowup. We highlight the comprehensiveness of our approach as we analyze numerous combinations of the energy and mass terms, despite the complexity involved. Interestingly, we find that when initial energy is minimized, classifications based on different functionals coincide. For high initial energy, an unresolved issue even for the standard Schrödinger equation, we develop a strategy controlling bounds of energy to classify the solution dynamics, including global existence or blowup, for our model. We also provide a complete proof for the standard case addressing this open problem.
Related
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- An inverse scattering problem for the nonlinear Schrödinger equation under partial harmonic confinement
- Blow-up of damped semilinear wave equation with short range potential and critical power nonlinearity
- Stable determination of damping and potential coefficients in a semilinear wave equation
- Determination of separable perturbations of an unbounded potential in the two-dimensional Schrödinger equation
BibTeX
@misc{han2026semilinear,
title = {{Semilinear Schrödinger equation with harmonic potential and combined power-type nonlinearities}},
author = {Jiangbo Han and Runzhang Xu and Chao Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32274},
eprint = {2610.32274},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32274},
}
RIS
TY - UNPB
TI - Semilinear Schrödinger equation with harmonic potential and combined power-type nonlinearities
AU - Jiangbo Han
AU - Runzhang Xu
AU - Chao Yang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32274
UR - https://arxiv.org/abs/2610.32274
AB - We study the Cauchy problem for the semilinear Schrödinger equation with a harmonic potential and combined power-type nonlinearities. The goal is to classify initial data that lead to either global existence or blowup solutions by using an enhanced cross-constrained variational theory. First, we review and analyze this variational approach, pinpointing the main issue for our specific problem. We then introduce new variational structures by carefully selecting terms in the energy functional and adjusting their weights, enabling us to create eight cross-constrained variational combinations and related manifolds to distinguish initial data that result in global solutions or finite-time blowup. Using these structures, we examine three initial energy levels: negative, non-negative low, and high energy. Notably, we prove that for any initial data with negative energy, solutions blow up in finite time regardless of their size. Throughout the study, we utilize these new variational frameworks and functionals to systematically classify initial data for global existence or finite-time blowup. We highlight the comprehensiveness of our approach as we analyze numerous combinations of the energy and mass terms, despite the complexity involved. Interestingly, we find that when initial energy is minimized, classifications based on different functionals coincide. For high initial energy, an unresolved issue even for the standard Schrödinger equation, we develop a strategy controlling bounds of energy to classify the solution dynamics, including global existence or blowup, for our model. We also provide a complete proof for the standard case addressing this open problem.
ER -