We consider the critical nonlinear Schrödinger equation $i{u}_{t}=-\Delta u-{\left|u\right|}^{\frac{4}{N}}u$ with initial condition $u(0,x)={u}_{0}$ in dimension $N$. For ${u}_{0}\in {H}^{1}$, local existence in time of solutions on an interval $[0,T)$ is known, and there exists finite time blow up solutions, that is ${u}_{0}$ such that ${lim}_{t\to T<+\infty}{\left|{u}_{x}\left(t\right)\right|}_{{L}^{2}}=+\infty $. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data known to lead to blow up, no general understanding of the blow up dynamic is known. At first, we propose in this paper a general setting to study and understand small in a certain sense blow up solutions. Blow up in finite time follows for the whole class of initial data in ${H}^{1}$ with strictly negative energy, and one is able to prove a control from above of the blow up rate below the one of the known explicit explosive solution, which has strictly positive energy.

@article{JEDP_2002____A12_0, author = {Merle, Frank and Raphael, Pierre}, title = {Blow up dynamic and upper bound on the blow up rate for critical nonlinear {Schr\"odinger} equation}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {12}, publisher = {Universit\'e de Nantes}, year = {2002}, doi = {10.5802/jedp.610}, mrnumber = {1968208}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jedp.610/} }

TY - JOUR AU - Merle, Frank AU - Raphael, Pierre TI - Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation JO - Journées équations aux dérivées partielles PY - 2002 DA - 2002/// PB - Université de Nantes UR - http://archive.numdam.org/articles/10.5802/jedp.610/ UR - https://www.ams.org/mathscinet-getitem?mr=1968208 UR - https://doi.org/10.5802/jedp.610 DO - 10.5802/jedp.610 LA - en ID - JEDP_2002____A12_0 ER -

%0 Journal Article %A Merle, Frank %A Raphael, Pierre %T Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation %J Journées équations aux dérivées partielles %D 2002 %I Université de Nantes %U https://doi.org/10.5802/jedp.610 %R 10.5802/jedp.610 %G en %F JEDP_2002____A12_0

Merle, Frank; Raphael, Pierre. Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation. Journées équations aux dérivées partielles (2002), article no. 12, 5 p. doi : 10.5802/jedp.610. http://archive.numdam.org/articles/10.5802/jedp.610/

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