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\textbf{Abstract}:\\
We study the blowup of  solutions to the Cauchy problem for  the $L^{2}$-critical damped nonlinear Schr\"{o}dinger equations (NLSa):
\begin{equation}\label{NLSa}
\left\{\begin{array}{l}
%$$ \left\lbrace \begin{array}{l}
iu_{t} + \Delta{u} +|u|^{\frac{4}{d}}u + iau =0,  (t,x) \in [0,\infty[\times {R}^{d}, d=1,2,3,4. \\
u(0)= u_{0} \in H^1({R}^{d})
%\end{array} $$
\end{array}\right.
\end{equation}
with initial data $u(0)= u_{0} \in H^1({R}^{d})$ and where $a > 0$ is the coefficient of friction.

 The blow up was expected in view of numerical simulations but not proven.  The available results on
the blowing up for  NLSa equations which are due to M. Tsutsumi
and more recently to Ohta-Todorova, treated only the super-critical case($|u|^{p}u$ with $p > \frac{4}{d}$)
by adapting the virial method. This does not seem to work in the critical
case and the idea is to prove that the log-log blowup regime described by
Merle-Raphael is stable by such perturbation of the equation.\\

\medskip
1. Darwich Mohamad. Blowup for the Damped $L^{2}$-Critical Nonlinear Schrodinger Equation. 
Advances in Differential Equations Volume 17, Numbers 3-4 (2012), 337--367.
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