Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS

Abstract

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i{\partial }_{t}u+\Delta u+k(x)|u{|}^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\Vert u{\Vert }_{L^2}<M_k$ are global in time while a finite time blow up singularity formation may occur for $\Vert u{\Vert }_{L^2}>M_k$. In this paper, we consider the dynamics at threshold $\Vert u_0{\Vert }_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow up elements at a non degenerate point, hence extending the pioneering work by Merle who treated the pseudo conformal invariant case $k\equiv 1$.

Cited by in corpus (1)

• Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation