Energy critical fourth-order Schrödinger equation system with power-type nonlinearities in the radial case
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915479447142400 |
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| author | Hespanha, Maicon Scarpelli, Renzo |
| author_facet | Hespanha, Maicon Scarpelli, Renzo |
| contents | In this paper, we study a system of focusing fourth-order Schrödinger equations in the energy-critical setting with radial initial data and general power-type nonlinearities. The main idea is to generalize the analysis of such systems: we first establish several hypotheses on the nonlinearities and prove their implications. These implications are then used to establish a local well-posedness result in $H^2(\mathbb{R}^d)$ and and to prove the existence of ground state solutions. Using a virial argument, we demonstrate a blow-up result for initial data with negative energy or with kinetic energy exceeding that of the ground state. Finally, employing the concentration-compactness/rigidity method, we prove a scattering result for solutions whose energy and kinetic energy are below those of the ground state. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Energy critical fourth-order Schrödinger equation system with power-type nonlinearities in the radial case Hespanha, Maicon Scarpelli, Renzo Analysis of PDEs 35Q44, 35P25, 35B44 In this paper, we study a system of focusing fourth-order Schrödinger equations in the energy-critical setting with radial initial data and general power-type nonlinearities. The main idea is to generalize the analysis of such systems: we first establish several hypotheses on the nonlinearities and prove their implications. These implications are then used to establish a local well-posedness result in $H^2(\mathbb{R}^d)$ and and to prove the existence of ground state solutions. Using a virial argument, we demonstrate a blow-up result for initial data with negative energy or with kinetic energy exceeding that of the ground state. Finally, employing the concentration-compactness/rigidity method, we prove a scattering result for solutions whose energy and kinetic energy are below those of the ground state. |
| title | Energy critical fourth-order Schrödinger equation system with power-type nonlinearities in the radial case |
| topic | Analysis of PDEs 35Q44, 35P25, 35B44 |
| url | https://arxiv.org/abs/2509.04428 |