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Divergent solutions to the 5D Hartree equations

Daomin CaoQing Guo — 2011

Colloquium Mathematicae

We consider the Cauchy problem for the focusing Hartree equation i u t + Δ u + ( | · | - 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 + Δ Q + ( | · | - 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 → ∞....

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