Divergent solutions to the 5D Hartree equations

Daomin Cao, and Qing Guo. "Divergent solutions to the 5D Hartree equations." Colloquium Mathematicae 125.2 (2011): 255-287. <http://eudml.org/doc/283488>.

@article{DaominCao2011,
abstract = {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.},
author = {Daomin Cao, Qing Guo},
journal = {Colloquium Mathematicae},
keywords = {Hartree equation; blow-up; profile decomposition; divergent solutions},
language = {eng},
number = {2},
pages = {255-287},
title = {Divergent solutions to the 5D Hartree equations},
url = {http://eudml.org/doc/283488},
volume = {125},
year = {2011},
}

TY - JOUR
AU - Daomin Cao
AU - Qing Guo
TI - Divergent solutions to the 5D Hartree equations
JO - Colloquium Mathematicae
PY - 2011
VL - 125
IS - 2
SP - 255
EP - 287
AB - 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.
LA - eng
KW - Hartree equation; blow-up; profile decomposition; divergent solutions
UR - http://eudml.org/doc/283488
ER -