Divergent solutions to the 5D Hartree equations

Daomin Cao¹ ; Qing Guo¹

¹ Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190, P.R. China

Colloquium Mathematicum, Tome 125 (2011) no. 2, pp. 255-287

Résumé

We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+\varDelta u+(|\cdot|^{-3}\ast|u|^{2})u=0$ in $\mathbb{R}^{5}$ with initial data in $H^1$, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of $-Q+\varDelta Q+(|\cdot|^{-3}\ast|Q|^{2})Q=0 $ in $ \mathbb{R}^{5}$, we prove that if $u_{0}\in H^{1}$ satisfies $M(u_0) E(u_0) M(Q) E(Q)$ and $\|\nabla u_{0}\|_{2}\|u_{0}\|_{2} >\|\nabla Q\|_{2}\|Q\|_{2} ,$ 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 $t_{n}\rightarrow\infty$ such that $\|\nabla u(t_{n})\|_{2}\rightarrow\infty.$ A similar result holds for negative time.

Détail

DOI : 10.4064/cm125-2-10

Mots-clés : consider cauchy problem focusing hartree equation vardelta cdot ast mathbb initial study divergence property infinite variance nonradial solutions ground state solution q vardelta cdot ast mathbb prove satisfies nabla nabla corresponding solution either blows finite forward time exists globally positive time there exists time sequence rightarrow infty nabla rightarrow infty similar result holds negative time