Divergent solutions to the 5D Hartree equations

• Autorzy: Daomin Cao , Qing Guo
• Języki publikacji: EN

Abstrakty

EN

We consider the Cauchy problem for the focusing Hartree equation $iu_{t} + Δu + (|·|^{-3} ∗ |u|²)u = 0$ in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of $-Q + ΔQ + (|·|^{-3} ∗ |Q|²)Q = 0$ in ℝ⁵, we prove that if u₀ ∈ H¹ satisfies M(u₀)E(u₀) < M(Q)E(Q) and ||∇u₀||₂||u₀||₂ > ||∇Q||₂||Q||₂, then the corresponding solution u(t) either blows up in finite forward time, or exists globally for positive time and there exists a time sequence tₙ → ∞ such that ||∇u(tₙ)||₂ → ∞. A similar result holds for negative time.

Kategorie tematyczne

• 35A15: Variational methods
• 35Q55: NLS-like equations (nonlinear Schr\"odinger)
• 35B30: Dependence of solutions on initial and boundary data, parameters

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2011
• Tom: 125
• Numer: 2
• Strony: 255-287

Daty

wydano

2011

Twórcy

autor

Daomin Cao

• Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China

autor

Qing Guo

• Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China

Identyfikatory

DOI

10.4064/cm125-2-10