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Divergent solutions to the 5D Hartree equations

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Cao D., Guo Q.

Colloquium Mathematicum

2011

tom 125

nr 2

255-287

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.

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1 Colloquium Mathematicum

1 Cao D.

1 Guo Q.

1 2011

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Divergent solutions to the 5D Hartree equations