Sharp threshold of blow-up and scattering for the fractional Hartree equation

By Qing Guo, Shihui Zhu. 2017

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Abstract We consider the fractional Hartree equation in the L 2 -supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If M [ u 0 ] s − s c s c E [ u 0 ] M [ Q ] s − s c s c E [ Q ] and M [ u 0 ] s − s c s c ‖ u 0 ‖ H ˙ s 2 M [ Q ] s − s c s c ‖ Q ‖ H ˙ s 2 , then the solution u ( t ) is globally well-posed and scatters; if M [ u 0 ] s − s c s c E [ u 0 ] M [ Q ] s − s c s c E [ Q ] and M [ u 0 ] s − s c s c ‖ u 0 ‖ H ˙ s 2 > M [ Q ] s − s c s c ‖ Q ‖ H ˙ s 2 , the solution u ( t ) blows up in finite time. This condition is sharp in the sense that the solitary wave solution e i t Q ( x ) is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d90e9ae94c3e2ab1d9d9bf1c639dec8a6facafe3