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

By Pierre Raphael, Jérémie Szeftel. 2010

143 citations recorded by the source corpus.

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+\Delta u+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2} M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ 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 $k\equiv 1$.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1842c7451c4da7b67bd88d906ed8819201c0bbef

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Blowup ofH1solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation (2017, background): “Merle in [21] and Raphaël-Szeftel in [24] studied (1.”
On well posedness for the inhomogeneous nonlinear Schr\"odinger equation (2016, methodology): “This model has been investigated by several authors, see, for instance, Merle [24] and Raphaël-Szeftel [25], for k1 < K(x) < k2 with k1, k2 > 0, and Fibich-Wang [11], for K(ǫ|x”
Blow up for the critical gKdV equation II: minimal mass dynamics (2012, methodology): “This is a delicate problem and here we adapt the direct dynamical approach developed in [52].”
On collapsing ring blow up solutions to the mass supercritical NLS (2012, methodology): “Following [37], [39], [31], the proof of (1.”
Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton (2014, methodology): “For this, we adapt and revisit the construction of mixed energy/virial functionals as introduced in [24], [39], [41] and [42].”