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Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS
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Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS
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Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS
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Journal Article
Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS
2011
Overview
We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: i∂tu+Δu+k(x)|u|2u=0i\\partial _tu+\\Delta u+k(x)|u|^{2}u=0. From a standard argument, there exists a threshold Mk>0M_k>0 such that H1H^1 solutions with ‖u‖L2>Mk\\|u\\|_{L^2}>M_k are global in time while a finite time blow-up singularity formation may occur for ‖u‖L2>Mk\\|u\\|_{L^2}>M_k. In this paper, we consider the dynamics at threshold ‖u0‖L2=Mk\\|u_0\\|_{L^2}=M_k and give a necessary and sufficient condition on kk 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 nondegenerate point, hence extending the pioneering work by Merle who treated the pseudoconformal invariant case k≡1k\\equiv 1.
Publisher
American Mathematical Society
Subject
